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First of all, I must clarify that I do not really know how much it is known about this subject, but, as it seems to be just a little bit/nothing (at least because of the little information I could find about it over the Internet), I will post this question anyway. Excuse me if you find it slightly "open".

Do we know any Ordinary Generating Function

$$f(x) = \sum_{n=0}^\infty a_n x^n$$

with some type of closed form for which $a_n$ contains or involves the Prime Counting Function $\pi(n)$?

To clarify, some examples of what I am looking for are:

$$f_1(x) = \sum_{n=0}^\infty \pi(n) x^n$$

$$f_2(x) = \sum_{n=0}^\infty \frac{\pi(n)}{n!} x^n$$

$$f_3(x) = \sum_{n=0}^\infty \frac{1}{\pi(n)} x^n$$

Etc.

Motivation: I thought about this while reading some conclusions about the famous formula

$$\int_2^\infty \frac{\pi(t)}{t^{s+1}-t} dt =\frac{\log \zeta(s)}{s}$$

For $\text{Re}(s)>1$.

And I asked myself whether we knew about any similar result on Ordinary Generating Functions.

Personally, it seems out of reach finding such a closed form. However, has anyone ever discovered any? It would be great if I could be referred to any paper about this kind of series, since I have not found any.

Thank you.

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1 Answer 1

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Let $\displaystyle \Psi(x) = \sum_{p^k} \frac{e^{-x p^k}}{k}$ the sum being over prime powers. Then for $\Re(s) > 1$ $$\int_0^\infty \Psi(x) x^{s-1}dx = \sum_{p^k} \frac{1}{k}\int_0^\infty e^{-x p^k} x^{s-1}dx=\sum_{p^k} \frac{1}{k} p^{-sk} \Gamma(s) = \Gamma(s) \log \zeta(s)$$ Thus by inverse Mellin/Laplace/Fourier transform, for $\sigma > 1$ $$\Psi(x) = \frac{1}{2i\pi} \int_{\sigma-i\infty}^{\sigma+i\infty} \frac{x^{-s}}{s}\Gamma(s)\log \zeta(s) ds$$ And the prime number theorem is that $\displaystyle\Psi(x) \sim \sum_{n=2}^\infty \frac{e^{-nx}}{\log n}$ as $x \to 0$.

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  • $\begingroup$ Thank you, but, on the one hand, that function is directly related to Prime Numbers, and I was searching for some series involving the Prime Counting Function as-is. On the other hand, I would not like to work with approaches by using the PNT, but with exact results of Ordinary Generating Functions. Thanks anyway $\endgroup$
    – user3141592
    Commented Aug 8, 2017 at 11:33
  • $\begingroup$ @user3141592 You need to think more : what I wrote is the answer. $\sum_p e^{-px}$ is directly related to $\sum_n \pi(n) e^{-nx}$ by summation by parts. Now the primes are very complicated : they are defined by the sieve of Erathosthenes. Only the Euler product is capable to give a simple relation between a (complicated) function encoding the primes and a (simple, $\zeta(s)$) function encoding the integers. $\endgroup$
    – reuns
    Commented Aug 8, 2017 at 11:36
  • $\begingroup$ Sorry, but I think that I do not get the point. The only generating function in your answer is $\displaystyle \Psi(x) = \sum_{p^k} \frac{e^{-x p^k}}{k}$, which is neither an ordinary generating function nor involves the Prime Counting Function. How would summation by parts solve these two problems? $\endgroup$
    – user3141592
    Commented Aug 8, 2017 at 11:43
  • $\begingroup$ @user3141592 It is an ordinary generating function and it involves the prime counting function. $\Psi(x) = \sum_n (\Pi(n)-\Pi(n-1)) e^{-nx} = \sum_n \Pi(n) e^{-nx} (1-e^{-x})$ where $\Pi(x) = \sum_{p^k \le x} \frac{1}{k}= \sum_m \pi(x^{1/k})$. $\endgroup$
    – reuns
    Commented Aug 8, 2017 at 11:45
  • $\begingroup$ @user3141592 anything unclear about why it answers your question ? $\endgroup$
    – reuns
    Commented Aug 8, 2017 at 12:24

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