Dedekind psi function

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The Dedekind psi function is defined by the formula

$\psi (n)=n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1+{\frac {1}{p}}{\bigg )}}$

or equivalently

$\psi (n)=\prod _{p^{\alpha }\parallel k}(p^{\alpha }+p^{\alpha -1})$

where as usual $p^{\alpha }\parallel n$ if $p^{\alpha }|n$ and $p^{\alpha +1}\not |n.$

Compare with Euler's totient function

$\varphi (n)=n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1-{\frac {1}{p}}{\bigg )}}$

Properties

The Dedekind psi function is a multiplicative arithmetic function, e.g.,

$\psi (mn)=\psi (m)\cdot \psi (n)$

when $gcd(m,n)=1.$

All values of the Dedekind psi function for $\scriptstyle n\geq 3$ are even.

Formulae

$\psi (n)\,\equiv \,n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1+{\frac {1}{p}}{\bigg )}}=n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\frac {\sigma _{1}(p)}{p}}}={\frac {n}{{\rm {rad}}(n)}}\cdot {\prod _{p|{\rm {rad}}(n) \atop p~{\rm {prime}}}{\sigma _{1}(p)}}={\frac {n}{{\rm {rad}}(n)}}\cdot {\sigma _{1}({\rm {rad}}(n))}={\frac {n}{{\rm {rad}}(n)}}\cdot {\psi ({\rm {rad}}(n))}$

where $\scriptstyle \sigma _{1}(n)$ is the sum of divisors of $\scriptstyle n$ and $\scriptstyle {\rm {rad}}(n)$ is the radical (squarefree kernel) of $\scriptstyle n$.

Generating function

Dirichlet generating function

$D_{\{\psi (n)\}}(s)\equiv \sum _{n=1}^{\infty }{\frac {\psi (n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-1)}{\zeta (2s)}}={\frac {(\zeta (s))^{2}}{\zeta (2s)}}D_{\{\varphi (n)\}}(s)$

Related functions

Dedekind psi summatory function

The Dedekind psi summatory function (partial sums of the Dedekind psi function) is

$\Psi (n)\equiv \sum _{k=1}^{n}\psi (k),$

where $\scriptstyle \psi (n)$ is Dedekind psi function.

All values of the Dedekind psi summatory function for $\scriptstyle n\geq 2$ are even.

Half difference of Dedekind psi function and Euler's totient function

${\frac {\psi (n)-\varphi (n)}{2}}={\frac {n}{2}}\cdot {{\bigg \{}{\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1+{\frac {1}{p}}{\bigg )}}-{\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1-{\frac {1}{p}}{\bigg )}}{\bigg \}}}$

n                1, 2, 3, 4, 5,  6, 7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, ...
psi(n)           1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, ...
phi(n)           1, 1, 2, 2, 4,  2, 6,  4,  6,  4, 10,  4, 12,  6,  8,  8, 16,  6, 18,  8, 12, 10, 22,  8, 20, 12, 18, 12, 28,  8, 30, 16, 20, 16, 24, 12, ...
psi(n) - phi(n)  0, 2, 2, 4, 2, 10, 2,  8,  6, 14,  2, 20,  2, 18, 16, 16,  2, 30,  2, 28, 20, 26,  2, 40, 10, 30, 18, 36,  2, 64,  2, 32, 28, 38, 24, 60, ... 
(psi - phi)/2    0, 1, 1, 2, 1,  5, 1,  4,  3,  7,  1, 10,  1,  9,  8,  8,  1, 15,  1, 14, 10, 13,  1, 20,  5, 15,  9, 18,  1, 32,  1, 16, 14, 19, 12, 30, ...
A069359          0, 1, 1, 2, 1,  5, 1,  4,  3,  7,  1, 10,  1,  9,  8,  8,  1, 15,  1, 14, 10, 13,  1, 20,  5, 15,  9, 18,  1,*31,* 1, 16, 14, 19, 12, 30, ...
A003415          0, 1, 1, 4, 1,  5, 1, 12,  6,  7,  1, 16,  1,  9,  8, 32,  1, 21,  1, 24, 10, 13,  1, 44, 10, 15, 27, 32,  1, 31,  1, 80, 14, 19, 12, 60, ...

The sequence for $\scriptstyle {\frac {\psi (n)-\varphi (n)}{2}}$ is nearly identical, at least for small $\scriptstyle n$, but is NOT A069359!

The above data suggests that $\scriptstyle {\frac {\psi (n)-\varphi (n)}{2}}$ is 0 only for $\scriptstyle n\,=\,1$ and 1 only when $\scriptstyle n$ is prime. It also suggests that when $\scriptstyle n$ is a prime power $\scriptstyle p^{k}$ that we get $\scriptstyle p^{k-1}$ in the sequence.

Product of the Dedekind psi function with Euler's totient function

The product of the Dedekind psi function with Euler's totient function gives

$\psi (n)\cdot \varphi (n)=n^{2}\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg \{}{\bigg (}1+{\frac {1}{p}}{\bigg )}{\bigg (}1-{\frac {1}{p}}{\bigg )}{\bigg \}}}=n^{2}\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1-{\frac {1}{p^{2}}}{\bigg )}}=n^{2}\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\frac {p^{2}-1}{p^{2}}}}={\bigg (}{\frac {n}{{\rm {rad}}(n)}}{\bigg )}^{2}\cdot {\prod _{p|n \atop p~{\rm {prime}}}(p^{2}-1)}$

where $\scriptstyle {\rm {rad}}(n)$ is the squarefree kernel of $\scriptstyle n$.

Since $\scriptstyle p^{2}-1$ is divisible by 24 (A024702) when $\scriptstyle p$ is congruent to 1 or 5 modulo 6 and $\scriptstyle (2^{2}-1)(3^{2}-1)\,=\,24$, we deduce that $\scriptstyle \psi (n)\cdot \varphi (n)$ is divisible by $\scriptstyle 24^{\omega (n)}$ if $\scriptstyle n$ is divisible by neither 2 nor 3 or both 2 and 3, and is divisible by $\scriptstyle 24^{\omega (n)-1}$ if $\scriptstyle n$ is divisible by either (but not both) 2 or 3, $\scriptstyle \omega (n)$ being the number of distinct prime factors of $\scriptstyle n$.

Also the Jordan function J_2(n) (a generalization of phi(n))

$\psi (n)\cdot \varphi (n)=J_{2}(n)$ (A007434)

Also the Moebius transform of squares

$\psi (n)\cdot \varphi (n)=J_{2}(n)=\sum _{d|n}\mu ({\tfrac {n}{d}})\cdot d^{2}$

Multiplicative with

$J_{2}(p^{e})=p^{2e}-p^{2e-2}=p^{2e}{\bigg (}1-{\frac {1}{p^{2}}}{\bigg )}$.

Quotient of the Dedekind psi function by Euler's totient function

The quotient of the Dedekind psi function by Euler's totient function gives

${\frac {\psi (n)}{\varphi (n)}}={\prod _{p|n \atop p~{\rm {prime}}}{\frac {{\bigg (}1+{\frac {1}{p}}{\bigg )}}{{\bigg (}1-{\frac {1}{p}}{\bigg )}}}}={\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}{\frac {p+1}{p-1}}{\bigg )}}$

Sequences

A001615 Dedekind psi function

ψ (n)

.

{1, 3, 4, 6, 6, 12, 8, 12, 12, 18, 12, 24, 14, 24, 24, 24, 18, 36, 20, 36, 32, 36, 24, 48, 30, 42, 36, 48, 30, 72, 32, 48, 48, 54, 48, 72, 38, 60, 56, 72, 42, 96, 44, 72, 72, 72, 48, 96, 56, 90, 72, 84, 54, 108, 72, 96, 80, ...}

A173290 Dedekind psi summatory function.

{1, 4, 8, 14, 20, 32, 40, 52, 64, 82, 94, 118, 132, 156, 180, 204, 222, 258, 278, 314, 346, 382, 406, 454, 484, 526, 562, 610, 640, 712, 744, 792, 840, 894, 942, 1014, 1052, 1112, 1168, 1240, 1282, 1378, 1422, ...}

A?????? (Dedekind ψ (n)  −  Euler’s totient φ (n))  / 2 (is nearly identical, at least for small

n

, but is NOT A069359!)

{0, 1, 1, 2, 1, 5, 1, 4, 3, 7, 1, 10, 1, 9, 8, 8, 1, 15, 1, 14, 10, 13, 1, 20, 5, 15, 9, 18, 1, 32, 1, 16, 14, 19, 12, 30, ...}

A007434 Jordan function

J2(n) = ψ (n)  ⋅  φ (n)

.

{1, 3, 8, 12, 24, 24, 48, 48, 72, 72, 120, 96, 168, 144, 192, 192, 288, 216, 360, 288, 384, 360, 528, 384, 600, 504, 648, 576, 840, 576, 960, 768, 960, 864, 1152, 864, 1368, 1080, 1344, 1152, 1680, 1152, 1848, ...}

See also

• {{Dedekind psi}} arithmetic function template

• Dedekind psi summatory function
• Euler's totient function
• Totient summatory function
• Euler's cototient function
• A007434 Jordan function J_2(n) (a generalization of phi(n)).

External links

• Weisstein, Eric W., Dedekind Function, from MathWorld—A Wolfram Web Resource.
• Jordan's totient function, from PlanetMath Encyclopedia—Math for the people, by the people.
• Dedekind psi function—Wikipedia.org.
• Jordan's totient function—Wikipedia.org.