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In number theory, the Dedekind psi function is the multiplicative function on the positive integers defined by

ψ ψ ( n ) = n ∏ ∏ p | n ( 1 + 1 p ) , {\displaystyle \psi (n)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right),}

where the product is taken over all primes p {\displaystyle p} dividing n . {\displaystyle n.} (By convention, ψ ψ ( 1 ) {\displaystyle \psi (1)} , which is the empty product, has value 1.) The function was introduced by Richard Dedekind in connection with modular functions.

The value of ψ ψ ( n ) {\displaystyle \psi (n)} for the first few integers n {\displaystyle n} is:

1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... (sequence A001615 in the OEIS).

The function ψ ψ ( n ) {\displaystyle \psi (n)} is greater than n {\displaystyle n} for all n {\displaystyle n} greater than 1, and is even for all n {\displaystyle n} greater than 2. If n {\displaystyle n} is a square-free number then ψ ψ ( n ) = σ σ ( n ) {\displaystyle \psi (n)=\sigma (n)} , where σ σ ( n ) {\displaystyle \sigma (n)} is the sum-of-divisors function.

The ψ ψ {\displaystyle \psi } function can also be defined by setting ψ ψ ( p n ) = ( p + 1 ) p n − − 1 {\displaystyle \psi (p^{n})=(p+1)p^{n-1}} for powers of any prime p {\displaystyle p} , and then extending the definition to all integers by multiplicativity. This also leads to a proof of the generating function in terms of the Riemann zeta function, which is

∑ ∑ ψ ψ ( n ) n s = ζ ζ ( s ) ζ ζ ( s − − 1 ) ζ ζ ( 2 s ) . {\displaystyle \sum {\frac {\psi (n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-1)}{\zeta (2s)}}.}

This is also a consequence of the fact that we can write as a Dirichlet convolution of ψ ψ = I d ∗ ∗ | μ μ | {\displaystyle \psi =\mathrm {Id} *|\mu |} .

There is an additive definition of the psi function as well. Quoting from Dickson,cite-ref-1[1]

R. Dedekindcite-ref-2[2] proved that, if n {\displaystyle n} is decomposed in every way into a product a b {\displaystyle ab} and if e {\displaystyle e} is the g.c.d. of a , b {\displaystyle a,b} then ∑ ∑ a ( a / e ) φ φ ( e ) = n ∏ ∏ p | n ( 1 + 1 p ) {\displaystyle \sum _{a}(a/e)\varphi (e)=n\prod _{p|n}\left(1+{\frac {1}{p}}\right)} where a {\displaystyle a} ranges over all divisors of n {\displaystyle n} and p {\displaystyle p} over the prime divisors of n {\displaystyle n} and φ φ {\displaystyle \varphi } is the totient function.

• Higher orders

• References

• External links

• See also

The generalization to higher orders via ratios of Jordan's totient is

ψ ψ k ( n ) = J 2 k ( n ) J k ( n ) {\displaystyle \psi _{k}(n)={\frac {J_{2k}(n)}{J_{k}(n)}}}

with Dirichlet series

∑ ∑ n ≥ ≥ 1 ψ ψ k ( n ) n s = ζ ζ ( s ) ζ ζ ( s − − k ) ζ ζ ( 2 s ) {\displaystyle \sum _{n\geq 1}{\frac {\psi _{k}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-k)}{\zeta (2s)}}} .

It is also the Dirichlet convolution of a power and the square of the Möbius function,

ψ ψ k ( n ) = n k ∗ ∗ μ μ 2 ( n ) {\displaystyle \psi _{k}(n)=n^{k}*\mu ^{2}(n)} .

If

ϵ ϵ 2 = 1 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 … … {\displaystyle \epsilon _{2}=1,0,0,1,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0\ldots }

is the characteristic function of the squares, another Dirichlet convolution leads to the generalized σ-function,

ϵ ϵ 2 ( n ) ∗ ∗ ψ ψ k ( n ) = σ σ k ( n ) {\displaystyle \epsilon _{2}(n)*\psi _{k}(n)=\sigma _{k}(n)} .

cite-note-11. ↑ Leonard Eugene Dickson "History of the Theory Of Numbers", Vol. 1, p. 123, Chelsea Publishing 1952.

cite-note-22. ↑ Journal für die reine und angewandte Mathematik, vol. 83, 1877, p. 288. Cf. H. Weber, Elliptische Functionen, 1901, 244-5; ed. 2, 1008 (Algebra III), 234-5

• reference-mathworld-dedekind-functionciterefweissteinWeisstein, Eric W. "Dedekind Function". MathWorld.

• citerefgoro-shimura1971Goro Shimura (1971). Introduction to the Arithmetic Theory of Automorphic Functions. Princeton. (page 25, equation (1))

• citerefmathar2011Mathar, Richard J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038 [math.NT]. Section 3.13.2

• OEIS: A065958 is ψ2, OEIS: A065959 is ψ3, and OEIS: A065960 is ψ4