It is well-known that the Euler $\phi$-function is multiplicative: that is, for co-prime positive integers $m,n$ we have $\phi(mn) = \phi(m)\phi(n)$. Thus it is defined by its values on prime powers. We know that $\phi(p^k) = p^{k-1} (p-1)$ for all primes $p$.

What about the multiplicative function $\psi$ defined on the primes by $\psi(p^k) = p^{k-1} (p+1)$? Does it have a name? If so, what's known about it?

For example, can one evaluate $\sum_{n \leq X} \psi(n)$?

up vote 12 down vote accepted

$\psi$ is the multiplicative convolution of $\mu^2$ and the identity function, hence its Dirichlet series is $$\sum_{n=1}^\infty\frac{\psi(n)}{n^s}=\frac{\zeta(s)\zeta(s-1)}{\zeta(2s)},\qquad\Re(s)>2.$$ This implies by Perron's formula and standard bounds that $$\sum_{n \leq X} \psi(n)\sim\frac{\zeta(2)}{2\zeta(4)}X^2=\frac{15}{2\pi^2}X^2.$$

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    I will just add this function appears in Exercise 11 in Chapter 3 in Apostol's book. The last part of this exercise is the asymptotic formula $\sum_{n\le x} \varphi_1(n)=\frac{\zeta(2)}{2\zeta(4)}x^2+O(x\log x)$. (Apostol denotes the function by $\varphi_1$.) – Martin Sleziak Oct 18 at 13:58
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    In the field of automorphic forms, this function is often denotes $\nu(n)$ (for example, in Iwaniec-Luo-Sarnak's seminal paper on low-lying zeroes of $L$-functions of holomorphic cuspidal eigenforms). – Peter Humphries Oct 18 at 14:07
  • @MartinSleziak: Thanks for your comment. Clearly, one can also prove this statement by elementary means, by making use of the convolution I mentioned. – GH from MO Oct 18 at 14:35
  • @PeterHumphries: Indeed the function looked familiar! It is also the index of $\Gamma_0(n)$ in $\Gamma_0(1)$ if I am not mistaken. – GH from MO Oct 18 at 14:36
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    @GHfromMO, yep, which is why it comes up in the field automorphic forms so often! – Peter Humphries Oct 18 at 15:03

This is also called the Dedekind $\psi$ function: $$ \psi(n):=n\prod_{p|n}(1+p^{-1}) $$

See also A001615 and A158523.

  • This should be a comment in my opinion, but thanks for it anyways. (Usually answers should be true answers, which really answer the original question.) – GH from MO Oct 18 at 14:38
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    @GHfromMO You obviously know more about the etiquette of this site than me, but I do believe I answered the main two questions in the OP: "does this have a name", and "what is known about it". I didn't say anything about the asymptotics, because that was already covered by you, and because that seemed to me as an example question, and not the main one. But I appreciate the feedback anyway. – AccidentalFourierTransform Oct 18 at 14:42
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    I agree with you. I forgot about those parts of the question! (BTW it is usually discouraged to ask several questions in one post, exactly to avoid confusions like this.) – GH from MO Oct 18 at 14:44
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    I find it amazing that I used the same letter as the generally accepted one! – Stanley Yao Xiao Oct 18 at 15:09

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