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Dirichlet convolution

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In <a href="Mathematics" title="Mathematics">mathematics</a>, Dirichlet convolution (or divisor convolution) is a <a href="Binary_operation" title="Binary operation">binary operation</a> defined for <a href="Arithmetic_function" title="Arithmetic function">arithmetic functions</a>; it is important in <a href="Number_theory" title="Number theory">number theory</a>. It was developed by <a href="Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g:\mathbb {N} \to \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle f,g:\mathbb {N} \to \mathbb {C} }</annotation>

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</math><img src="./dda43a5747c9ff1df761d8acacadb46de059a9dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.336ex; height:2.509ex;" alt="{\displaystyle f,g:\mathbb {N} \to \mathbb {C} }" loading="lazy"> are two <a href="Arithmetic_function" title="Arithmetic function">arithmetic functions</a>, their Dirichlet convolution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}">

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<annotation encoding="application/x-tex">{\displaystyle f*g}</annotation>

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</math><img src="./de088e4a3777d3b5d2787fdec81acd91e78a719e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}" loading="lazy"> is a new arithmetic function defined by:

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<annotation encoding="application/x-tex">{\displaystyle (f*g)(n)\ =\ \sum _{d\,\mid \,n}f(d)\,g\!\left({\frac {n}{d}}\right)\ =\ \sum _{ab\,=\,n}\!f(a)\,g(b),}</annotation>

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</math><img src="./d2aa49f5878a4c46064a6786a53e4be5830715fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:47.727ex; height:6.509ex;" alt="{\displaystyle (f*g)(n)\ =\ \sum _{d\,\mid \,n}f(d)\,g\!\left({\frac {n}{d}}\right)\ =\ \sum _{ab\,=\,n}\!f(a)\,g(b),}" loading="lazy"></dd></dl>

where the sum extends over all positive <a href="Divisor" title="Divisor">divisors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

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</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, or equivalently over all distinct pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)}">

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<annotation encoding="application/x-tex">{\displaystyle (a,b)}</annotation>

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</math><img src="./d7e5710198f33b00695903460983021e75860e2c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.071ex; height:2.843ex;" alt="{\displaystyle (a,b)}" loading="lazy"> of positive integers whose product is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">.

This product occurs naturally in the study of <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a> such as the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. It describes the multiplication of two Dirichlet series in terms of their coefficients:

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<annotation encoding="application/x-tex">{\displaystyle \left(\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}\right)\left(\sum _{n\geq 1}{\frac {g(n)}{n^{s}}}\right)\ =\ \left(\sum _{n\geq 1}{\frac {(f*g)(n)}{n^{s}}}\right).}</annotation>

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</math><img src="./26018ae5020608199f0adf555f89670c797c23ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.639ex; height:7.509ex;" alt="{\displaystyle \left(\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}\right)\left(\sum _{n\geq 1}{\frac {g(n)}{n^{s}}}\right)\ =\ \left(\sum _{n\geq 1}{\frac {(f*g)(n)}{n^{s}}}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

The set of arithmetic functions forms a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>, the <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>Dirichlet ring, with addition given by <a href="Pointwise_addition" class="mw-redirect" title="Pointwise addition">pointwise addition</a> and multiplication by Dirichlet convolution. The multiplicative identity is the <a href="Unit_function" title="Unit function">unit function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (n)=1}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon (n)=1}</annotation>

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</math><img src="./809584f727cc795f634a5edad6f4e87e9202dc89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.548ex; height:2.843ex;" alt="{\displaystyle \varepsilon (n)=1}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

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<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

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</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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<annotation encoding="application/x-tex">{\displaystyle 0}</annotation>

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</math><img src="./2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> otherwise. The <a href="Unit_(ring_theory)" title="Unit (ring theory)">units</a> (invertible elements) of this ring are the arithmetic functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)\neq 0}">

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<annotation encoding="application/x-tex">{\displaystyle f(1)\neq 0}</annotation>

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</math><img src="./5339d843bc8ee5244cc90a45de9832a657c04645.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(1)\neq 0}" loading="lazy">.

Specifically, Dirichlet convolution is <a href="Associativity" class="mw-redirect" title="Associativity">associative</a>,<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle (f*g)*h=f*(g*h),}</annotation>

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</math><img src="./d4ee094f2e0af3f9754977794b5952fc0b9e9dc0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.61ex; height:2.843ex;" alt="{\displaystyle (f*g)*h=f*(g*h),}" loading="lazy"></dd></dl>

<a href="Distributivity" class="mw-redirect" title="Distributivity">distributive</a> over addition

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<annotation encoding="application/x-tex">{\displaystyle f*(g+h)=f*g+f*h}</annotation>

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</math><img src="./43b4420e33d10761c8ea395538ed2cc1509e957a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.918ex; height:2.843ex;" alt="{\displaystyle f*(g+h)=f*g+f*h}" loading="lazy">,</dd></dl>

<a href="Commutativity" class="mw-redirect" title="Commutativity">commutative</a>,

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<annotation encoding="application/x-tex">{\displaystyle f*g=g*f}</annotation>

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</math><img src="./611ab4d408ca1fd090dd1e2647d9583aafa80e63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.277ex; height:2.509ex;" alt="{\displaystyle f*g=g*f}" loading="lazy">,</dd></dl>

and has an identity element,

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<annotation encoding="application/x-tex">{\displaystyle f*\varepsilon }</annotation>

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</math><img src="./509d10f0c180fbb816c0004f0e8562c782be5d87.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.557ex; height:2.509ex;" alt="{\displaystyle f*\varepsilon }" loading="lazy"> = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon *f=f}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon *f=f}</annotation>

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</math><img src="./d16b2fbcddac9bde2a24ce389dedd4082f41865f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.934ex; height:2.509ex;" alt="{\displaystyle \varepsilon *f=f}" loading="lazy">.</dd></dl>

Furthermore, for each function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> having <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)\neq 0}">

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</math><img src="./5339d843bc8ee5244cc90a45de9832a657c04645.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(1)\neq 0}" loading="lazy">, there exists another arithmetic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}}</annotation>

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</math><img src="./3e5cfa2f5c08d6fe7d046b73faa6e3f213acc802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.653ex; height:3.009ex;" alt="{\displaystyle f^{-1}}" loading="lazy"> satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*f^{-1}=\varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle f*f^{-1}=\varepsilon }</annotation>

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</math><img src="./6e83d854f39b5204a80f9bae9871c4e824674e59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.309ex; height:3.009ex;" alt="{\displaystyle f*f^{-1}=\varepsilon }" loading="lazy">, called the Dirichlet inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.

The Dirichlet convolution of two <a href="Multiplicative_function" title="Multiplicative function">multiplicative functions</a> is again multiplicative, and every not constantly zero multiplicative function has a Dirichlet inverse which is also multiplicative. In other words, multiplicative functions form a subgroup of the group of invertible elements of the Dirichlet ring. Beware however that the sum of two multiplicative functions is not multiplicative (since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(1)=f(1)+g(1)=2\neq 1}">

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<annotation encoding="application/x-tex">{\displaystyle (f+g)(1)=f(1)+g(1)=2\neq 1}</annotation>

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</math><img src="./801e18fc097ba0a97f0be93672e1de10db4f2871.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.815ex; height:2.843ex;" alt="{\displaystyle (f+g)(1)=f(1)+g(1)=2\neq 1}" loading="lazy">), so the subset of multiplicative functions is not a subring of the Dirichlet ring. The article on multiplicative functions lists several convolution relations among important multiplicative functions.

Another operation on arithmetic functions is pointwise multiplication: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg}">

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<annotation encoding="application/x-tex">{\displaystyle fg}</annotation>

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</math><img src="./06bac4638bb56f14688118ce88c188c7a021eb29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle fg}" loading="lazy"> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (fg)(n)=f(n)g(n)}">

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<annotation encoding="application/x-tex">{\displaystyle (fg)(n)=f(n)g(n)}</annotation>

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</math><img src="./5c2eace46bbd5c34b72082b17e5ad369162348a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.309ex; height:2.843ex;" alt="{\displaystyle (fg)(n)=f(n)g(n)}" loading="lazy">. Given a <a href="Completely_multiplicative_function" title="Completely multiplicative function">completely multiplicative function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">, pointwise multiplication by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

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<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

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</math><img src="./b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> distributes over Dirichlet convolution: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)h=(fh)*(gh)}">

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<annotation encoding="application/x-tex">{\displaystyle (f*g)h=(fh)*(gh)}</annotation>

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</math><img src="./2c7ad446bf9157b969a23b234df89ca499092859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.722ex; height:2.843ex;" alt="{\displaystyle (f*g)h=(fh)*(gh)}" loading="lazy">.<a href="#cite_note-2">[2]</a> The convolution of two completely multiplicative functions is multiplicative, but not necessarily completely multiplicative.

<div class="mw-heading mw-heading2"><h2 id="Properties_and_examples">Properties and examples</h2></div>

In these formulas, we use the following <a href="Arithmetical_function" class="mw-redirect" title="Arithmetical function">arithmetical functions</a>:

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> is the multiplicative identity: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (1)=1}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon (1)=1}</annotation>

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</math><img src="./5a5b64bd390c8eb2f16f6c30fe34150ccdf00813.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.316ex; height:2.843ex;" alt="{\displaystyle \varepsilon (1)=1}" loading="lazy">, otherwise 0 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon (n)=\lfloor {\tfrac {1}{n}}\rfloor }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon (n)=\lfloor {\tfrac {1}{n}}\rfloor }</annotation>

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</math><img src="./702904c0ed73985f8cb9c4699c5cca897d5b70fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.273ex; height:3.343ex;" alt="{\displaystyle \varepsilon (n)=\lfloor {\tfrac {1}{n}}\rfloor }" loading="lazy">).</li>

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> is the constant function with value 1: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1(n)=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1(n)=1}</annotation>

</semantics>

</math><img src="./882dc7f3b21154370c11c231caf7a3f7955facf2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.627ex; height:2.843ex;" alt="{\displaystyle 1(n)=1}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">. Keep in mind that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> is not the identity. (Some authors <a href="Incidence_algebra#Special_elements" title="Incidence algebra">denote this</a> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

</semantics>

</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"> because the associated Dirichlet series is the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>.)</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{C}}</annotation>

</semantics>

</math><img src="./c2a2fad618e7531ee3933e9f35e684d2f6f82a2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.644ex; height:2.509ex;" alt="{\displaystyle 1_{C}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\subset \mathbb {N} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C\subset \mathbb {N} }</annotation>

</semantics>

</math><img src="./c0773a88a1b3b08e695c5038c5a9be42ce447fe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.543ex; height:2.176ex;" alt="{\displaystyle C\subset \mathbb {N} }" loading="lazy"> is a set <a href="Indicator_function" title="Indicator function">indicator function</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{C}(n)=1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{C}(n)=1}</annotation>

</semantics>

</math><img src="./388d6eab6a5722cd2b4f6ceef7346fbab19cf98d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.109ex; height:2.843ex;" alt="{\displaystyle 1_{C}(n)=1}" loading="lazy"> iff <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\in C}</annotation>

</semantics>

</math><img src="./b43c827b0c5a0634a2cc28ba9c38f978764771df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.002ex; height:2.176ex;" alt="{\displaystyle n\in C}" loading="lazy">, otherwise 0.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}}</annotation>

</semantics>

</math><img src="./0406ec36a5bc81001681058dcf97fa21833191ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.132ex; height:2.176ex;" alt="{\displaystyle {\text{Id}}}" loading="lazy"> is the identity function with value n: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Id}}(n)=n}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}(n)=n}</annotation>

</semantics>

</math><img src="./7f9d9f9c690e3373f160b3ee72d0749c08542b03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.829ex; height:2.843ex;" alt="{\displaystyle {\text{Id}}(n)=n}" loading="lazy">.</li>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}_{k}}</annotation>

</semantics>

</math><img src="./b0a4e3aad6f757ca1f0577d780039e8f8a0576e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.221ex; height:2.509ex;" alt="{\displaystyle {\text{Id}}_{k}}" loading="lazy"> is the kth power function: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{Id}}_{k}(n)=n^{k}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}_{k}(n)=n^{k}}</annotation>

</semantics>

</math><img src="./80c82e548de9cc91cc7144d648786bbc46583e41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.007ex; height:3.176ex;" alt="{\displaystyle {\text{Id}}_{k}(n)=n^{k}}" loading="lazy">.</li></ul>

The following relations hold:

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 1*\mu =\varepsilon }</annotation>

</semantics>

</math><img src="./355fe362ab5f4ece14eca0cf863d10d67d4c1049.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.941ex; height:2.676ex;" alt="{\displaystyle 1*\mu =\varepsilon }" loading="lazy">, the Dirichlet inverse of the constant function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

</semantics>

</math><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"> is the <a href="M%C3%B6bius_function" title="Möbius function">Möbius function</a> (see <a href="M%C3%B6bius_function#Proof_of_the_formula_for_the_sum_of_μ_over_divisors" title="Möbius function">proof</a>). Hence:</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle g=f*1}</annotation>

</semantics>

</math><img src="./814f332b6d1b54c1ea58c8ca26b040e6ad0619f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.85ex; height:2.509ex;" alt="{\displaystyle g=f*1}" loading="lazy"> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=g*\mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=g*\mu }</annotation>

</semantics>

</math><img src="./76dbcde47a7f596590ca5222af55b1d55b755d1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.089ex; height:2.676ex;" alt="{\displaystyle f=g*\mu }" loading="lazy">, the <a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion formula</a>.</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{k}={\text{Id}}_{k}*1}</annotation>

</semantics>

</math><img src="./c8ca2fbf6a82a102762e3fd6ace6869329bbdc14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.093ex; height:2.509ex;" alt="{\displaystyle \sigma _{k}={\text{Id}}_{k}*1}" loading="lazy">, the <a href="Divisor_function" title="Divisor function">kth-power-of-divisors sum function</a> σk.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma ={\text{Id}}*1}</annotation>

</semantics>

</math><img src="./ee395165c4fe25abc4483fe5955ad4bc66458dd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.917ex; height:2.176ex;" alt="{\displaystyle \sigma ={\text{Id}}*1}" loading="lazy">, the sum-of-divisors function σ = σ1.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau =1*1}</annotation>

</semantics>

</math><img src="./5c9504d4210f0d9a04d9e2e8f4723e87a10c6367.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.82ex; height:2.176ex;" alt="{\displaystyle \tau =1*1}" loading="lazy"> , the number-of-divisors function τ(n) = σ0.</li>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}_{k}=\sigma _{k}*\mu }</annotation>

</semantics>

</math><img src="./fdb9c637d5d8e114b4eb6b7ad102bf6b2cba1728.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.332ex; height:2.676ex;" alt="{\displaystyle {\text{Id}}_{k}=\sigma _{k}*\mu }" loading="lazy">,  by Möbius inversion of the formulas for σk, σ, and τ.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}=\sigma *\mu }</annotation>

</semantics>

</math><img src="./136c53c7c493f6aa511e76d567d13a59c6aefba8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.157ex; height:2.676ex;" alt="{\displaystyle {\text{Id}}=\sigma *\mu }" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1=\tau *\mu }</annotation>

</semantics>

</math><img src="./e7a9b8e9b4964bf085f31117cd52827a25e10f1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.059ex; height:2.676ex;" alt="{\displaystyle 1=\tau *\mu }" loading="lazy"></li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi *1={\text{Id}}}</annotation>

</semantics>

</math><img src="./2c4ac9ffce75025a26093761d4e662a6a0a3a197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.973ex; height:2.509ex;" alt="{\displaystyle \phi *1={\text{Id}}}" loading="lazy"> , proved under <a href="Euler's_totient_function#Divisor_sum" title="Euler's totient function">Euler's totient function</a>.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi ={\text{Id}}*\mu }</annotation>

</semantics>

</math><img src="./caef1d26f269f8bcaada15f6fb3297eef0108d94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.212ex; height:2.676ex;" alt="{\displaystyle \phi ={\text{Id}}*\mu }" loading="lazy"> , by Möbius inversion.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma =\phi *\tau }</annotation>

</semantics>

</math><img src="./b42f38d858fe9de66a2a4aeaeddddbebc8da6a02.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.21ex; height:2.509ex;" alt="{\displaystyle \sigma =\phi *\tau }" loading="lazy">  , from convolving 1 on both sides of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi *1={\text{Id}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi *1={\text{Id}}}</annotation>

</semantics>

</math><img src="./2c4ac9ffce75025a26093761d4e662a6a0a3a197.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.973ex; height:2.509ex;" alt="{\displaystyle \phi *1={\text{Id}}}" loading="lazy">.</li>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda *|\mu |=\varepsilon }</annotation>

</semantics>

</math><img src="./9b1b69cbf1d25afdc727a8a4361e1cae9a333375.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.427ex; height:2.843ex;" alt="{\displaystyle \lambda *|\mu |=\varepsilon }" loading="lazy">  where λ is <a href="Liouville's_function" class="mw-redirect" title="Liouville's function">Liouville's function</a>.</li>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}*\phi =P}</annotation>

</semantics>

</math><img src="./5373e756fa4b9abc1d7d01f0295a80d7f1e51ab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.556ex; height:2.509ex;" alt="{\displaystyle {\text{Id}}*\phi =P}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> is <a href="Pillai's_arithmetical_function" title="Pillai's arithmetical function">Pillai's arithmetical function</a>, also known as the gcd-sum function.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda *1=1_{\text{Sq}}}</annotation>

</semantics>

</math><img src="./2dcf8a47158e3a9a17208df83c55ce7d4886d0b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.987ex; height:2.843ex;" alt="{\displaystyle \lambda *1=1_{\text{Sq}}}" loading="lazy"> where Sq = {1, 4, 9, ...} is the set of squares.</li>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{Id}}_{k}*({\text{Id}}_{k}\mu )=\varepsilon }</annotation>

</semantics>

</math><img src="./673f5a8af9fbd55562d805c54670be4def2b34d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.029ex; height:2.843ex;" alt="{\displaystyle {\text{Id}}_{k}*({\text{Id}}_{k}\mu )=\varepsilon }" loading="lazy"></li>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tau ^{3}*1=(\tau *1)^{2}}</annotation>

</semantics>

</math><img src="./37bb0f3f7f9146fee47fc7ea92c72fa9eca59643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.19ex; height:3.176ex;" alt="{\displaystyle \tau ^{3}*1=(\tau *1)^{2}}" loading="lazy"></li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{k}*1={\text{Id}}_{k}}</annotation>

</semantics>

</math><img src="./19a838dd9199bf67d73b9a8cd12c3d3d0a5da5fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.055ex; height:2.509ex;" alt="{\displaystyle J_{k}*1={\text{Id}}_{k}}" loading="lazy">, <a href="Jordan's_totient_function" title="Jordan's totient function">Jordan's totient function</a>.</li>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ({\text{Id}}_{s}J_{r})*J_{s}=J_{s+r}}</annotation>

</semantics>

</math><img src="./91da7b42d0095538ed9dbe4d4bc524ee965291c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.109ex; height:2.843ex;" alt="{\displaystyle ({\text{Id}}_{s}J_{r})*J_{s}=J_{s+r}}" loading="lazy"></li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda *1=\log }</annotation>

</semantics>

</math><img src="./ebab7aade297af072e9c6775a9bbc56557f3d4ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.04ex; height:2.509ex;" alt="{\displaystyle \Lambda *1=\log }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda }</annotation>

</semantics>

</math><img src="./0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy"> is <a href="Von_Mangoldt_function" title="Von Mangoldt function">von Mangoldt's function</a>.</li>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |\mu |\ast 1=2^{\omega },}</annotation>

</semantics>

</math><img src="./e0b7ed0cf004355d6d32375a3b6f08b826c6a28d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.215ex; height:2.843ex;" alt="{\displaystyle |\mu |\ast 1=2^{\omega },}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega (n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega (n)}</annotation>

</semantics>

</math><img src="./f9e1e190faf1ba7931e4e98594df6097d297bed1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.65ex; height:2.843ex;" alt="{\displaystyle \omega (n)}" loading="lazy"> is the <a href="Prime_omega_function" title="Prime omega function">prime omega function</a> counting distinct prime factors of n.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega \ast \mu =1_{\mathcal {P}}}</annotation>

</semantics>

</math><img src="./0632ea9ab635c15d647cba5efdfd861a4f96aad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.972ex; height:2.676ex;" alt="{\displaystyle \Omega \ast \mu =1_{\mathcal {P}}}" loading="lazy">, the characteristic function of the prime powers.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega \ast \mu =1_{\mathbb {P} }}</annotation>

</semantics>

</math><img src="./79674ca6546c1db8af8a418188f7b64648446842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.54ex; height:2.676ex;" alt="{\displaystyle \omega \ast \mu =1_{\mathbb {P} }}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{\mathbb {P} }(n)\mapsto \{0,1\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1_{\mathbb {P} }(n)\mapsto \{0,1\}}</annotation>

</semantics>

</math><img src="./93b83277be60bb1d1fb9cc98372e6438e49a8433.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.901ex; height:2.843ex;" alt="{\displaystyle 1_{\mathbb {P} }(n)\mapsto \{0,1\}}" loading="lazy"> is the characteristic function of the primes.</li></ul>

This last identity shows that the <a href="Prime-counting_function" title="Prime-counting function">prime-counting function</a> is given by the summatory function

</mrow>

</munder>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi (x)=\sum _{n\leq x}(\omega \ast \mu )(n)=\sum _{d=1}^{x}\omega (d)M\left(\left\lfloor {\frac {x}{d}}\right\rfloor \right)}</annotation>

</semantics>

</math><img src="./ef76e2eab6f77d92891ca5089f56cf032d289fb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.517ex; height:7.009ex;" alt="{\displaystyle \pi (x)=\sum _{n\leq x}(\omega \ast \mu )(n)=\sum _{d=1}^{x}\omega (d)M\left(\left\lfloor {\frac {x}{d}}\right\rfloor \right)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M(x)}</annotation>

</semantics>

</math><img src="./8d2fab18f2df9b523ef8b7b63a291317294fc708.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.581ex; height:2.843ex;" alt="{\displaystyle M(x)}" loading="lazy"> is the <a href="Mertens_function" title="Mertens function">Mertens function</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> is the distinct prime factor counting function from above. This expansion follows from the identity for the sums over Dirichlet convolutions given on the <a href="Divisor_sum_identities" title="Divisor sum identities">divisor sum identities</a> page (a standard trick for these sums).<a href="#cite_note-3">[3]</a>

<div class="mw-heading mw-heading2"><h2 id="Dirichlet_inverse">Dirichlet inverse</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Examples">Examples</h3></div>

Given an arithmetic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> its Dirichlet inverse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=f^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g=f^{-1}}</annotation>

</semantics>

</math><img src="./87a0b4704a13840b49bb0addc39e11bde430debe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.868ex; height:3.009ex;" alt="{\displaystyle g=f^{-1}}" loading="lazy"> may be calculated recursively: the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(n)}</annotation>

</semantics>

</math><img src="./d4ad18070e494503403daf39398e711c1378348e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\displaystyle g(n)}" loading="lazy"> is in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(m)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(m)}</annotation>

</semantics>

</math><img src="./b6c16e07bfd8831a41d15a9830d0b32abacf0c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.966ex; height:2.843ex;" alt="{\displaystyle g(m)}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m<n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m<n}</annotation>

</semantics>

</math><img src="./490c01b0cb770144f28afd17bb5fef277daf6f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle m<n}" loading="lazy">.

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=1}</annotation>

</semantics>

</math><img src="./d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}" loading="lazy">:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)(1)=f(1)g(1)=\varepsilon (1)=1}</annotation>

</semantics>

</math><img src="./c50fc5f373e87800950d4f10fe14bf9623939c1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.222ex; height:2.843ex;" alt="{\displaystyle (f*g)(1)=f(1)g(1)=\varepsilon (1)=1}" loading="lazy">, so</dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(1)=1/f(1)}</annotation>

</semantics>

</math><img src="./00e0461c75bcb13e981ca496a00754b50703f03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.761ex; height:2.843ex;" alt="{\displaystyle g(1)=1/f(1)}" loading="lazy">. This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> does not have a Dirichlet inverse if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(1)=0}</annotation>

</semantics>

</math><img src="./da761295ab324acdb04e33ab75299c7fb588675f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(1)=0}" loading="lazy">.</dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2}</annotation>

</semantics>

</math><img src="./a02c8bd752d2cc859747ca1f3a508281bdbc3b34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=2}" loading="lazy">:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)(2)=f(1)g(2)+f(2)g(1)=\varepsilon (2)=0}</annotation>

</semantics>

</math><img src="./3be6a371254da11dd59de25170a6d8c943f8de84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.4ex; height:2.843ex;" alt="{\displaystyle (f*g)(2)=f(1)g(2)+f(2)g(1)=\varepsilon (2)=0}" loading="lazy">,</dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(2)=-(f(2)g(1))/f(1)}</annotation>

</semantics>

</math><img src="./2fcd75061b1b488659a4713a94523f22bf85cfc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.555ex; height:2.843ex;" alt="{\displaystyle g(2)=-(f(2)g(1))/f(1)}" loading="lazy">,</dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=3}</annotation>

</semantics>

</math><img src="./1c5a5a42ced00df920fad4ab2d4acdb960a4105b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=3}" loading="lazy">:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)(3)=f(1)g(3)+f(3)g(1)=\varepsilon (3)=0}</annotation>

</semantics>

</math><img src="./6cb2f80b6dbdc29c7c5a9159c161261a5956b9d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.4ex; height:2.843ex;" alt="{\displaystyle (f*g)(3)=f(1)g(3)+f(3)g(1)=\varepsilon (3)=0}" loading="lazy">,</dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(3)=-(f(3)g(1))/f(1)}</annotation>

</semantics>

</math><img src="./8bb3f6801a547e85556c36dbae5903849bb85530.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.555ex; height:2.843ex;" alt="{\displaystyle g(3)=-(f(3)g(1))/f(1)}" loading="lazy">,</dd></dl>

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=4}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=4}</annotation>

</semantics>

</math><img src="./d928ec15aeef83aade867992ee473933adb6139d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=4}" loading="lazy">:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f*g)(4)=f(1)g(4)+f(2)g(2)+f(4)g(1)=\varepsilon (4)=0}</annotation>

</semantics>

</math><img src="./5972c73d0bf1cc3f86edeae7c88c21a52c199a62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.579ex; height:2.843ex;" alt="{\displaystyle (f*g)(4)=f(1)g(4)+f(2)g(2)+f(4)g(1)=\varepsilon (4)=0}" loading="lazy">,</dd>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(4)=-(f(4)g(1)+f(2)g(2))/f(1)}</annotation>

</semantics>

</math><img src="./1d5fb2bbfa581cc14b3523dfb8f350defee1e014.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.733ex; height:2.843ex;" alt="{\displaystyle g(4)=-(f(4)g(1)+f(2)g(2))/f(1)}" loading="lazy">,</dd></dl>

and in general for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>1}</annotation>

</semantics>

</math><img src="./ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}" loading="lazy">,

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

</mrow>

</mrow>

</munder>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(n)\ =\ {\frac {-1}{f(1)}}\mathop {\sum _{d\,\mid \,n}} _{d<n}f\left({\frac {n}{d}}\right)g(d).}</annotation>

</semantics>

</math><img src="./b8b6cb2b0edb5f6e30d71aa8b4cb5992f9468723.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:29.643ex; height:8.676ex;" alt="{\displaystyle g(n)\ =\ {\frac {-1}{f(1)}}\mathop {\sum _{d\,\mid \,n}} _{d<n}f\left({\frac {n}{d}}\right)g(d).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Properties_2">Properties</h3></div>

The following properties of the Dirichlet inverse hold:<a href="#cite_note-4">[4]</a>

<ul><li>The function f has a Dirichlet inverse if and only if f(1) ≠ 0.</li>

<li>The Dirichlet inverse of a <a href="Multiplicative_function" title="Multiplicative function">multiplicative function</a> is again multiplicative.</li>

<li>The Dirichlet inverse of a Dirichlet convolution is the convolution of the inverses of each function: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\ast g)^{-1}=f^{-1}\ast g^{-1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f\ast g)^{-1}=f^{-1}\ast g^{-1}}</annotation>

</semantics>

</math><img src="./3dbf9f56bcb704fa726f3c407ee1626f8f125bd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.129ex; height:3.176ex;" alt="{\displaystyle (f\ast g)^{-1}=f^{-1}\ast g^{-1}}" loading="lazy">.</li>

<li>A multiplicative function f is <a href="Completely_multiplicative" class="mw-redirect" title="Completely multiplicative">completely multiplicative</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(n)=\mu (n)f(n)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(n)=\mu (n)f(n)}</annotation>

</semantics>

</math><img src="./c4be579806aa606ba7f6814d174ca19b331a95f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.044ex; height:3.176ex;" alt="{\displaystyle f^{-1}(n)=\mu (n)f(n)}" loading="lazy">.</li>

<li>If f is <a href="Completely_multiplicative" class="mw-redirect" title="Completely multiplicative">completely multiplicative</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\cdot g)^{-1}=f\cdot g^{-1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f\cdot g)^{-1}=f\cdot g^{-1}}</annotation>

</semantics>

</math><img src="./a7d70bc3efa13319a58538dac3f5b41c19ec5d9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.723ex; height:3.176ex;" alt="{\displaystyle (f\cdot g)^{-1}=f\cdot g^{-1}}" loading="lazy"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(1)\neq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(1)\neq 0}</annotation>

</semantics>

</math><img src="./2d0e4d120ce89a4d006e557e65d0eb3b9243bebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.349ex; height:2.843ex;" alt="{\displaystyle g(1)\neq 0}" loading="lazy"> and where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cdot }</annotation>

</semantics>

</math><img src="./ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"> denotes pointwise multiplication of functions.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Other_formulas">Other formulas</h3></div>

<th>Arithmetic function</th>

<th>Dirichlet inverse:<a href="#cite_note-5">[5]</a>

</th></tr>

<tr>

<td>Constant function with value 1</td>

<td><a href="M%C3%B6bius_function" title="Möbius function">Möbius function</a> μ

</td></tr>

<tr>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n^{\alpha }}</annotation>

</semantics>

</math><img src="./ba0659b75ed58dffc477d189c59c09e229c81085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.679ex; height:2.343ex;" alt="{\displaystyle n^{\alpha }}" loading="lazy"></td>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu (n)\,n^{\alpha }}</annotation>

</semantics>

</math><img src="./fc3c07618fa16d58a00f3bab026d68f859fd2e63.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.672ex; height:2.843ex;" alt="{\displaystyle \mu (n)\,n^{\alpha }}" loading="lazy">

</td></tr>

<tr>

<td><a href="Liouville's_function" class="mw-redirect" title="Liouville's function">Liouville's function</a> λ</td>

<td>Absolute value of Möbius function |μ|

</td></tr>

<tr>

<td><a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>

</semantics>

</math><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></td>

</mrow>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{d|n}d\,\mu (d)}</annotation>

</semantics>

</math><img src="./204fdf9e8729ac5b86eb95c0134fa246fe937a51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:9.772ex; height:6.009ex;" alt="{\displaystyle \sum _{d|n}d\,\mu (d)}" loading="lazy">

</td></tr>

<tr>

<td>The <a href="Sum_of_divisors" class="mw-redirect" title="Sum of divisors">generalized sum-of-divisors function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\alpha }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{\alpha }}</annotation>

</semantics>

</math><img src="./0cc839b7526074471cc424b75b854cc6481e6074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.612ex; height:2.009ex;" alt="{\displaystyle \sigma _{\alpha }}" loading="lazy"></td>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{d|n}d^{\alpha }\mu (d)\mu \left({\frac {n}{d}}\right)}</annotation>

</semantics>

</math><img src="./cb5cb8ad00f3871a43747aa235bdcf851e80ede0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:17.466ex; height:6.509ex;" alt="{\displaystyle \sum _{d|n}d^{\alpha }\mu (d)\mu \left({\frac {n}{d}}\right)}" loading="lazy">

</td></tr></tbody></table>

An exact, non-recursive formula for the Dirichlet inverse of any <a href="Arithmetic_function" title="Arithmetic function">arithmetic function</a> f is given in <a href="Divisor_sum_identities#The_Dirichlet_inverse_of_an_arithmetic_function" title="Divisor sum identities">Divisor sum identities</a>. A more <a href="Partition_theory" class="mw-redirect" title="Partition theory">partition theoretic</a> expression for the Dirichlet inverse of f is given by

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mfrac>

</mrow>

</munder>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(n)=\sum _{k=1}^{\Omega (n)}\left\{\sum _{{\lambda _{1}+2\lambda _{2}+\cdots +k\lambda _{k}=n} \atop {\lambda _{1},\lambda _{2},\ldots ,\lambda _{k}|n}}{\frac {(\lambda _{1}+\lambda _{2}+\cdots +\lambda _{k})!}{1!2!\cdots k!}}(-1)^{k}f(\lambda _{1})f(\lambda _{2})^{2}\cdots f(\lambda _{k})^{k}\right\}.}</annotation>

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</math><img src="./c5df4397b6a487d92e079c0c79aef69859b48869.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:82.383ex; height:11.509ex;" alt="{\displaystyle f^{-1}(n)=\sum _{k=1}^{\Omega (n)}\left\{\sum _{{\lambda _{1}+2\lambda _{2}+\cdots +k\lambda _{k}=n} \atop {\lambda _{1},\lambda _{2},\ldots ,\lambda _{k}|n}}{\frac {(\lambda _{1}+\lambda _{2}+\cdots +\lambda _{k})!}{1!2!\cdots k!}}(-1)^{k}f(\lambda _{1})f(\lambda _{2})^{2}\cdots f(\lambda _{k})^{k}\right\}.}" loading="lazy"></dd></dl>

The following formula provides a compact way of expressing the Dirichlet inverse of an invertible arithmetic function f :

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}=\sum _{k=0}^{+\infty }{\frac {(f(1)\varepsilon -f)^{*k}}{f(1)^{k+1}}}}</annotation>

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</math><img src="./63cfcf8f2da93d6d298ff91487f77ba1a0d94c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.503ex; height:7.343ex;" alt="{\displaystyle f^{-1}=\sum _{k=0}^{+\infty }{\frac {(f(1)\varepsilon -f)^{*k}}{f(1)^{k+1}}}}" loading="lazy">

where the expression <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(1)\varepsilon -f)^{*k}}">

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<annotation encoding="application/x-tex">{\displaystyle (f(1)\varepsilon -f)^{*k}}</annotation>

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</math><img src="./48438ef60e84f92d3a6893485da6b88cf74cd7b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.173ex; height:3.176ex;" alt="{\displaystyle (f(1)\varepsilon -f)^{*k}}" loading="lazy"> stands for the arithmetic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)\varepsilon -f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(1)\varepsilon -f}</annotation>

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</math><img src="./c3677e912cb06755a36298bb9ab637c7d39fb094.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.453ex; height:2.843ex;" alt="{\displaystyle f(1)\varepsilon -f}" loading="lazy"> convoluted with itself k times. Notice that, for a fixed positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>\Omega (n)}">

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<annotation encoding="application/x-tex">{\displaystyle k>\Omega (n)}</annotation>

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</math><img src="./b1f524df9e8e65124fd7afb14efcd47094248b75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.192ex; height:2.843ex;" alt="{\displaystyle k>\Omega (n)}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f(1)\varepsilon -f)^{*k}(n)=0}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle (f(1)\varepsilon -f)^{*k}(n)=0}</annotation>

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</math><img src="./130393048b93ef147ca1862c9491cbb44632ec0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.638ex; height:3.176ex;" alt="{\displaystyle (f(1)\varepsilon -f)^{*k}(n)=0}" loading="lazy"> , this is because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)\varepsilon (1)-f(1)=0}">

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<annotation encoding="application/x-tex">{\displaystyle f(1)\varepsilon (1)-f(1)=0}</annotation>

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</math><img src="./17045cbd2a8d4850d42d3307fe92683d85c6e232.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.657ex; height:2.843ex;" alt="{\displaystyle f(1)\varepsilon (1)-f(1)=0}" loading="lazy"> and every way of expressing n as a product of k positive integers must include a 1, so the series on the right hand side converges for every fixed positive integer n.

<div class="mw-heading mw-heading2"><h2 id="Dirichlet_series">Dirichlet series</h2></div>

If f is an arithmetic function, the <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a> <a href="Generating_function" title="Generating function">generating function</a> is defined by

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<annotation encoding="application/x-tex">{\displaystyle DG(f;s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}}}</annotation>

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</math><img src="./6be1ab2e9cef9faa96b06d83700875f1820c8111.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.123ex; height:6.843ex;" alt="{\displaystyle DG(f;s)=\sum _{n=1}^{\infty }{\frac {f(n)}{n^{s}}}}" loading="lazy"></dd></dl>

for those <a href="Complex_number" title="Complex number">complex</a> arguments s for which the series converges (if there are any). The multiplication of Dirichlet series is compatible with Dirichlet convolution in the following sense:

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle DG(f;s)DG(g;s)=DG(f*g;s)\,}</annotation>

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</math><img src="./c5b7176c642416a9e9610fe3d5d2abe2274ee3d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.523ex; height:2.843ex;" alt="{\displaystyle DG(f;s)DG(g;s)=DG(f*g;s)\,}" loading="lazy"></dd></dl>

for all s for which both series of the left hand side converge, one of them at least converging

absolutely (note that simple convergence of both series of the left hand side does not imply convergence of the right hand side!). This is akin to the <a href="Convolution_theorem" title="Convolution theorem">convolution theorem</a> if one thinks of Dirichlet series as a <a href="Fourier_transform" title="Fourier transform">Fourier transform</a>.

<div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2></div>

The restriction of the divisors in the convolution to <a href="Unitary_divisor" title="Unitary divisor">unitary</a>, <a href="Bi-unitary_divisor" class="mw-redirect" title="Bi-unitary divisor">bi-unitary</a> or infinitary divisors defines similar commutative operations which share many features with the Dirichlet convolution (existence of a Möbius inversion, persistence of multiplicativity, definitions of totients, Euler-type product formulas over associated primes, etc.).

Dirichlet convolution is a special case of the convolution multiplication for the <a href="Incidence_algebra" title="Incidence algebra">incidence algebra</a> of a <a href="Partially_ordered_set" title="Partially ordered set">poset</a>, in this case the poset of positive integers ordered by divisibility.

The <a href="Dirichlet_hyperbola_method" title="Dirichlet hyperbola method">Dirichlet hyperbola method</a> computes the summation of a convolution in terms of its functions and their summation functions.

<ul><li><a href="Arithmetic_function" title="Arithmetic function">Arithmetic function</a></li>

<li><a href="Divisor_sum_identities" title="Divisor sum identities">Divisor sum identities</a></li>

<li><a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion formula</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Proofs are in Chan, ch. 2

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> A proof can be found in <a href="Completely_multiplicative_function#Proof_of_distributive_property" title="Completely multiplicative function">this article</a>.

</li>

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</style><cite id="CITEREFSchmidt" class="citation book cs1">Schmidt, Maxie. Apostol's Introduction to Analytic Number Theory.</cite> This identity is a little special something I call "croutons". It follows from several chapters worth of exercises in Apostol's classic book.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Again see Apostol Chapter 2 and the exercises at the end of the chapter.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> See Apostol Chapter 2.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFApostol1976" class="citation cs2"><a href="Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90163-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></cite></li>

<li><cite id="CITEREFChan,_Heng_Huat2009" class="citation book cs1">Chan, Heng Huat (2009). Analytic Number Theory for Undergraduates. Monographs in Number Theory. World Scientific Publishing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-4271-36-3</bdi>.</cite></li>

<li><cite id="CITEREFHugh_L._MontgomeryRobert_C._Vaughan2007" class="citation book cs1"><a href="Hugh_Montgomery_(mathematician)" class="mw-redirect" title="Hugh Montgomery (mathematician)">Hugh L. Montgomery</a>; <a href="Robert_Charles_Vaughan_(mathematician)" class="mw-redirect" title="Robert Charles Vaughan (mathematician)">Robert C. Vaughan</a> (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. Vol. 97. Cambridge: Cambridge Univ. Press. p. 38. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-84903-6</bdi>.</cite></li>

<li><cite id="CITEREFCohen1959" class="citation news cs1">Cohen, Eckford (1959). "A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion". Pacific J. Math. Vol. 9, no. 1. pp. 13–23. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0109806">0109806</a>.</cite></li>

<li><cite id="CITEREFCohen1960" class="citation journal cs1">Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". <a href="Mathematische_Zeitschrift" title="Mathematische Zeitschrift">Mathematische Zeitschrift</a>. 74: 66–80. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01180473">10.1007/BF01180473</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0112861">0112861</a>.</cite></li>

<li><cite id="CITEREFCohen1960" class="citation news cs1">Cohen, Eckford (1960). "The number of unitary divisors of an integer". <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. Vol. 67, no. 9. pp. 879–880. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0122790">0122790</a>.</cite></li>

<li><cite id="CITEREFCohen1990" class="citation journal cs1">Cohen, Graeme L. (1990). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1990-0993927-5">"On an integers' infinitary divisors"</a>. Math. Comp. 54 (189): 395–411. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0025-5718-1990-0993927-5">10.1090/S0025-5718-1990-0993927-5</a>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0993927">0993927</a>.</cite></li>

<li><cite id="CITEREFCohen1993" class="citation journal cs1">Cohen, Graeme L. (1993). <a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FS0161171293000456">"Arithmetic functions associated with infinitary divisors of an integer"</a>. Int. J. Math. Math. Sci. 16 (2): 373–383. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1155%2FS0161171293000456">10.1155/S0161171293000456</a>.</cite></li>

<li><cite id="CITEREFHaukkanen2000" class="citation journal cs1">Haukkanen, Pentti (2000). <a rel="nofollow" class="external text" href="https://nntdm.net/volume-06-2000/number-4/118-124/">"Expressions for the Dirichlet inverse of arithmetical functions"</a>. Notes on Number Theory and Discrete Mathematics. 6 (4): 118–124.</cite></li>

<li><cite id="CITEREFSandorBerge2003" class="citation journal cs1">Sandor, Jozsef; Berge, Antal (2003). "The Möbius function: generalizations and extensions". Adv. Stud. Contemp. Math. (Kyungshang). 6 (2): 77–128. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1962765">1962765</a>.</cite></li>

<li><cite id="CITEREFFinch2004" class="citation web cs1">Finch, Steven (2004). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150222094526/http://www.people.fas.harvard.edu/~sfinch/csolve/try.pdf">"Unitarism and Infinitarism"</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://www.people.fas.harvard.edu/~sfinch/csolve/try.pdf">the original</a> (PDF) on 2015-02-22.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Dirichlet_convolution">"Dirichlet convolution"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></li></ul>

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</style><div id="Peter_Gustav_Lejeune_Dirichlet34" style="font-size:114%;margin:0 4em"><a href="Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dirichlet_distribution" title="Dirichlet distribution">Dirichlet distribution</a></li>

<li><a href="Dirichlet_character" title="Dirichlet character">Dirichlet character</a></li>

<li><a href="Dirichlet_process" title="Dirichlet process">Dirichlet process</a></li>

<li><a href="Dirichlet-multinomial_distribution" title="Dirichlet-multinomial distribution">Dirichlet-multinomial distribution</a></li>

<li><a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a></li>

<li><a href="Dirichlet's_theorem_on_arithmetic_progressions" title="Dirichlet's theorem on arithmetic progressions">Dirichlet's theorem on arithmetic progressions</a></li>

<li><a href="Dirichlet_problem" title="Dirichlet problem">Dirichlet problem</a></li>

<li><a href="Dirichlet_integral" title="Dirichlet integral">Dirichlet integral</a></li></ul>

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