Dedekind's psi function

The Dedekind ψ function is one of several number theoretic functions named after Richard Dedekind . It's a multiplicative function , it's done

${\displaystyle \forall \ n\in \mathbb {Z} ^{+}\colon \psi (n)=n\cdot \prod _{p|n \atop p\in \mathbb {P} }\left(1+{\frac {1}{p}}\right)}$

Are defined. The product covers all prime divisors of ${\displaystyle n.}$

values

According to the definition of the empty product is

${\displaystyle \psi (1)=1.}$

For the next two natural numbers we get:

${\displaystyle \psi (2)=2\left(1+{\frac {1}{2}}\right)=3}$

${\displaystyle \psi (3)=3\left(1+{\frac {1}{3}}\right)=4}$

The sequence of function values ​​continues with 6, 6, 12, 8, 12, 12, 18, 12, 24,….

properties

• The function only takes positive natural numbers as values. For all sufficiently large is greater than and straight: ${\displaystyle \psi }$ ${\displaystyle n}$ ${\displaystyle \psi (n)}$ ${\displaystyle n}$

${\displaystyle \psi (n)>n\qquad \qquad \qquad \mathrm {f{\ddot {u}}r\;alle} \,n>1}$

${\displaystyle \psi (n)\equiv 0\mod 2\;\qquad \mathrm {f{\ddot {u}}r\;alle} \,n>2}$

• The following applies to prime numbers : ${\displaystyle p}$

${\displaystyle \psi (p)=p+1=\varphi (p)+2}$

It is the Euler's totient function , which for each positive integer , the number of the to prime indicating natural numbers not greater than are. ${\displaystyle \varphi }$ ${\displaystyle n}$ ${\displaystyle \varphi (n)}$ ${\displaystyle n}$ ${\displaystyle n}$

• The function can also be carried out by ${\displaystyle \psi }$

${\displaystyle \psi (p^{k})=(p+1)\cdot p^{k-1}}$

for powers of prime numbers with positive natural exponents and the definition that it is multiplicative. The value for any is then obtained from the prime factorization of ${\displaystyle p}$ ${\displaystyle k}$ ${\displaystyle \psi }$ ${\displaystyle \psi (n)}$ ${\displaystyle n}$ ${\displaystyle n.}$

• With the Riemann zeta function : ${\displaystyle \zeta }$

${\displaystyle \sum _{n}{\frac {\psi (n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-1)}{\zeta (2s)}}}$

Web links

• Eric W. Weisstein : Dedekind Function . In: MathWorld (English).
• J. Chidambaraswamy: Generalized Dedekind psi functions with respect to a polynomial. II. In: Pacific J. Math. Vol. 65, No. 1 (1976), pp. 19-27.

swell

1. ↑ Follow A001615 in OEIS