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The Dedekind psi function is defined by the formula
![{\displaystyle \psi (n)=n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1+{\frac {1}{p}}{\bigg )}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3fc7249830e0d287a4ce5f2e856d7bf1be19cb92)
or equivalently
![{\displaystyle \psi (n)=\prod _{p^{\alpha }\parallel k}(p^{\alpha }+p^{\alpha -1})}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/2bdfd477f4f18821cb1397c0bb41499303726837)
where as usual
if
and
Compare with Euler's totient function
![{\displaystyle \varphi (n)=n\cdot {\prod _{p|n \atop p~{\rm {prime}}}{\bigg (}1-{\frac {1}{p}}{\bigg )}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ca00029e85d12bd8e9c1b013ca62a133b8d81e62)
Properties
The Dedekind psi function is a multiplicative arithmetic function, e.g.,
![{\displaystyle \psi (mn)=\psi (m)\cdot \psi (n)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d1d69c9a6890ec699635d515719944bd713ac7b3)
when
All values of the Dedekind psi function for
are even.
Formulae
where
is the sum of divisors of
and
is the radical (squarefree kernel) of
.
Generating function
Dirichlet generating function
![{\displaystyle 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)}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/ccbc3bea065adabc4b9b43104d7bb4cecf149429)
Related functions
Dedekind psi summatory function
The Dedekind psi summatory function (partial sums of the Dedekind psi function) is
![{\displaystyle \Psi (n)\equiv \sum _{k=1}^{n}\psi (k),}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/7ebf26a39b8cd9e2a82282fafc7928364b60510b)
where
is Dedekind psi function.
All values of the Dedekind psi summatory function for
are even.
Half difference of Dedekind psi function and Euler's totient function
![{\displaystyle {\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 \}}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/eb7bea96173f3021b7fb394289b45390912e18f8)
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
is nearly identical, at least for small
, but is NOT A069359!
The above data suggests that
is 0 only for
and 1 only when
is prime. It also suggests that when
is a prime power
that we get
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
![{\displaystyle \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)}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/9d9242e965efa65aa4556e08ea451bf5ca796d17)
where
is the squarefree kernel of
.
Since
is divisible by 24 (A024702) when
is congruent to 1 or 5 modulo 6 and
, we deduce that
is divisible by
if
is divisible by neither 2 nor 3 or both 2 and 3, and is divisible by
if
is divisible by either (but not both) 2 or 3,
being the number of distinct prime factors of
.
Also the Jordan function J_2(n) (a generalization of phi(n))
(A007434)
Also the Moebius transform of squares
![{\displaystyle \psi (n)\cdot \varphi (n)=J_{2}(n)=\sum _{d|n}\mu ({\tfrac {n}{d}})\cdot d^{2}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/f978357342c36242ff6ac1c7e5506caea58c9f42)
Multiplicative with
.
Quotient of the Dedekind psi function by Euler's totient function
The quotient of the Dedekind psi function by Euler's totient function gives
![{\displaystyle {\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 )}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/76b7568e1625da496af4011dad8a6a8b8b0622ee)
Sequences
A001615 Dedekind psi function
.
-
{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
, 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
.
-
{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
External links