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Additive function

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</style><div role="note" class="hatnote navigation-not-searchable">For the <a href="Abstract_algebra" title="Abstract algebra">algebraic</a> meaning, see <a href="Additive_map" title="Additive map">Additive map</a>.</div>

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In <a href="Number_theory" title="Number theory">number theory</a>, an additive function is an <a href="Arithmetic_function" title="Arithmetic function">arithmetic function</a> f(n) of the positive <a href="Integer" title="Integer">integer</a> variable n such that whenever a and b are <a href="Coprime" class="mw-redirect" title="Coprime">coprime</a>, the function applied to the product ab is the sum of the values of the function applied to a and b:<a href="#cite_note-Erdos1939-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(ab)=f(a)+f(b).}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Completely_additive">Completely additive</h2></div>

An additive function f(n) is said to be completely additive if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(ab)=f(a)+f(b)}">

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

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</math><img src="./bc81dc1a82737a8c42b21f5ceca671d0036eccb3.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(ab)=f(a)+f(b)}" loading="lazy"> holds for all positive integers a and b, even when they are not coprime. Totally additive is also used in this sense by analogy with <a href="Totally_multiplicative" class="mw-redirect" title="Totally multiplicative">totally multiplicative</a> functions. If f is a completely additive function then f(1) = 0.

Every completely additive function is additive, but not vice versa.

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

Examples of arithmetic functions which are completely additive are:

<ul><li>The restriction of the <a href="Logarithm" title="Logarithm">logarithmic function</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} .}">

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

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</math><img src="./682f44bd6a1ea39ecf1e21a8290b9d5b2f504505.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.325ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} .}" loading="lazy"></li>

<li>The multiplicity of a <a href="Prime_number" title="Prime number">prime</a> factor p in n, that is the largest exponent m for which pm <a href="Divisor" title="Divisor">divides</a> n.</li>

<li> a0(n) – the sum of primes dividing n counting multiplicity, sometimes called sopfr(n), the potency of n or the integer logarithm of n (sequence <a href="https://oeis.org/A001414" class="extiw external" title="oeis:A001414">A001414</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). For example:</li></ul>

<ul><li>The function Ω(n), defined as the total number of <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> of n, counting multiple factors multiple times, sometimes called the "Big Omega function" (sequence <a href="https://oeis.org/A001222" class="extiw external" title="oeis:A001222">A001222</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). For example;</li></ul>

<dl><dd><dl><dd>Ω(1) = 0, since 1 has no prime factors</dd>

Examples of arithmetic functions which are additive but not completely additive are:

<ul><li>ω(n), defined as the total number of distinct <a href="Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> of n (sequence <a href="https://oeis.org/A001221" class="extiw external" title="oeis:A001221">A001221</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). For example:</li></ul>

<ul><li>a1(n) – the sum of the distinct primes dividing n, sometimes called sopf(n) (sequence <a href="https://oeis.org/A008472" class="extiw external" title="oeis:A008472">A008472</a> in the <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>). For example:</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Multiplicative_functions">Multiplicative functions</h2></div>

From any additive function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)}">

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

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</math><img src="./c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" loading="lazy"> it is possible to create a related <a href="Multiplicative_function" title="Multiplicative function">multiplicative function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n),}">

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

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</math><img src="./1ab472d6c6cbf538283f6ec4566e9300cc685ba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.967ex; height:2.843ex;" alt="{\displaystyle g(n),}" loading="lazy"> which is a function with the property that whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"> are coprime then:

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<annotation encoding="application/x-tex">{\displaystyle g(ab)=g(a)\times g(b).}</annotation>

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One such example is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)=2^{f(n)}.}">

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

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</math><img src="./1118b0a321fb11ef301f0603b897b3607ebe15a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.63ex; height:3.343ex;" alt="{\displaystyle g(n)=2^{f(n)}.}" loading="lazy"> Likewise if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)}">

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

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</math><img src="./c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}" loading="lazy"> is completely additive, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)=2^{f(n)}}">

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

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</math><img src="./ddd0f9fc694e959a2bf34ddd1aeb7443d6d9e8f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.983ex; height:3.343ex;" alt="{\displaystyle g(n)=2^{f(n)}}" loading="lazy"> is completely multiplicative. More generally, we could consider the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(n)=c^{f(n)}}">

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

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</math><img src="./78fe76947b062f2a644d2d65acb51e6a954514f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.827ex; height:3.343ex;" alt="{\displaystyle g(n)=c^{f(n)}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> is a nonzero real constant.

<div class="mw-heading mw-heading2"><h2 id="Summatory_functions">Summatory functions</h2></div>

Given an additive 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">, let its summatory function be defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)}">

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<annotation encoding="application/x-tex">{\textstyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)}</annotation>

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</math><img src="./fff977752c25c1c34ae131ecd3d2d27b889ba82d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.572ex; height:3.176ex;" alt="{\textstyle {\mathcal {M}}_{f}(x):=\sum _{n\leq x}f(n)}" loading="lazy">. The average 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"> is given exactly as

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{f}(x)=\sum _{p^{\alpha }\leq x}f(p^{\alpha })\left(\left\lfloor {\frac {x}{p^{\alpha }}}\right\rfloor -\left\lfloor {\frac {x}{p^{\alpha +1}}}\right\rfloor \right).}</annotation>

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The summatory functions over <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"> can be expanded as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))}</annotation>

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</math><img src="./2afd98ed840965bef6777ede060637dd900e1379.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.84ex; height:3.009ex;" alt="{\displaystyle {\mathcal {M}}_{f}(x)=xE(x)+O({\sqrt {x}}\cdot D(x))}" loading="lazy"> where

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E(x)&=\sum _{p^{\alpha }\leq x}f(p^{\alpha })p^{-\alpha }(1-p^{-1})\\D^{2}(x)&=\sum _{p^{\alpha }\leq x}|f(p^{\alpha })|^{2}p^{-\alpha }.\end{aligned}}}</annotation>

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The average of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{2}}">

<msup>

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

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</math><img src="./81efa211d9d04493b68b24bf9843d1967ae22cdc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:3.009ex;" alt="{\displaystyle f^{2}}" loading="lazy"> is also expressed by these functions as

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {M}}_{f^{2}}(x)=xE^{2}(x)+O(xD^{2}(x)).}</annotation>

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There is always an absolute constant <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{f}>0}">

<msub>

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

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</math><img src="./7326d35743452c13e39883cfdcf8ddb710336350.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.059ex; height:2.843ex;" alt="{\displaystyle C_{f}>0}" loading="lazy"> such that for all <a href="Natural_number" title="Natural number">natural numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\geq 1}">

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

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</math><img src="./5ca3ced43f1713577888a8a7ade2d0aaf8354a4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.591ex; height:2.343ex;" alt="{\displaystyle x\geq 1}" loading="lazy">,

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n\leq x}|f(n)-E(x)|^{2}\leq C_{f}\cdot xD^{2}(x).}</annotation>

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Let

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<annotation encoding="application/x-tex">{\displaystyle \nu (x;z):={\frac {1}{x}}\#\!\left\{n\leq x:{\frac {f(n)-A(x)}{B(x)}}\leq z\right\}\!.}</annotation>

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Suppose that <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"> is an additive function with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -1\leq f(p^{\alpha })=f(p)\leq 1}">

<msup>

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

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</math><img src="./87c909ce123950db24fb44e110257d9941a4378c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.227ex; height:2.843ex;" alt="{\displaystyle -1\leq f(p^{\alpha })=f(p)\leq 1}" loading="lazy">

such that as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\rightarrow \infty }">

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

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</math><img src="./02579e74e2ef1ca0befceba816b311fe5bfd6844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.268ex; height:1.843ex;" alt="{\displaystyle x\rightarrow \infty }" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle B(x)=\sum _{p\leq x}f^{2}(p)/p\rightarrow \infty .}</annotation>

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Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu (x;z)\sim G(z)}">

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<annotation encoding="application/x-tex">{\displaystyle \nu (x;z)\sim G(z)}</annotation>

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</math><img src="./8688dda9a20484c6201c08ab08f44ae6b4445b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.316ex; height:2.843ex;" alt="{\displaystyle \nu (x;z)\sim G(z)}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(z)}">

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

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</math><img src="./c56c297de4b4d6b42f3bdbfa47ffe1a81c45c147.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle G(z)}" loading="lazy"> is the <a href="Normal_distribution" title="Normal distribution">Gaussian distribution function</a>

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<annotation encoding="application/x-tex">{\displaystyle G(z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-t^{2}/2}dt.}</annotation>

</semantics>

</math>

Examples of this result related to the <a href="Prime_omega_function" title="Prime omega function">prime omega function</a> and the numbers of prime divisors of shifted primes include the following for fixed <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z\in \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z\in \mathbb {R} }</annotation>

</semantics>

</math><img src="./471dfb5d907062f413ab0eca669b4c8be4053a8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.607ex; height:2.176ex;" alt="{\displaystyle z\in \mathbb {R} }" loading="lazy"> where the relations hold for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\gg 1}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./03817e8d3d6b485b7304c4b481d7a352da1f4f0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.106ex; height:2.176ex;" alt="{\displaystyle x\gg 1}" loading="lazy">:

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \#\{n\leq x:\omega (n)-\log \log x\leq z(\log \log x)^{1/2}\}\sim xG(z),}</annotation>

</semantics>

</math>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \#\{p\leq x:\omega (p+1)-\log \log x\leq z(\log \log x)^{1/2}\}\sim \pi (x)G(z).}</annotation>

</semantics>

</math>

<ul><li><a href="Sigma_additivity" class="mw-redirect" title="Sigma additivity">Sigma additivity</a></li>

<li><a href="Prime_omega_function" title="Prime omega function">Prime omega function</a></li>

<li><a href="Multiplicative_function" title="Multiplicative function">Multiplicative function</a></li>

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

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

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<li id="cite_note-Erdos1939-1"><a href="#cite_ref-Erdos1939_1-0">^</a> Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207. <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077746/">online</a>

</li>

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

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

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<ul><li>Janko Bračič, Kolobar aritmetičnih funkcij (<a href="Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">Ring</a> of arithmetical functions), (Obzornik mat, fiz. 49 (2002) 4, pp. 97–108) (MSC (2000) 11A25) </li>

<li>Iwaniec and Kowalski, Analytic number theory, AMS (2004).</li></ul>

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