Dedekind psi function

{{Short description|Arithmetical function}}

{{Use dmy dates|date=July 2020}}

{{Use American English|date=July 2020}}

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

:$\backslash psi(n)\; =\; n\; \backslash prod\_\{p|n\}\backslash left(1+\backslash frac\{1\}\{p\}\backslash right),$

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

The value of $\backslash psi(n)$ for the first few integers $n$ is:

:1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, ... {{OEIS|id=A001615}}.

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

The $\backslash psi$ function can also be defined by setting $\backslash psi(p^n)\; =\; (p+1)p^\{n-1\}$ for powers of any prime $p$, 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

:$\backslash sum\; \backslash frac\{\backslash psi(n)\}\{n^s\}\; =\; \backslash frac\{\backslash zeta(s)\; \backslash zeta(s-1)\}\{\backslash zeta(2s)\}.$

This is also a consequence of the fact that we can write as a Dirichlet convolution of $\backslash psi=\; \backslash mathrm\{Id\}\; *\; |\backslash mu|$.

There is an additive definition of the psi function as well. Quoting from Dickson,Leonard Eugene Dickson "History of the Theory Of Numbers", Vol. 1, p. 123, Chelsea Publishing 1952.

R. DedekindJournal 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 proved that, if $n$ is decomposed in every way into a product $ab$ and if $e$ is the g.c.d. of $a,\; b$ then

:$\backslash sum\_\{a\}\; (a/e)\; \backslash varphi(e)\; =\; n\; \backslash prod\_\{p|n\}\backslash left(1+\backslash frac\{1\}\{p\}\backslash right)$

where $a$ ranges over all divisors of $n$ and $p$ over the prime divisors of $n$ and $\backslash varphi$ is the totient function.