Is there a "naturally occurring" function $f$ which is meromorphic in the complex plane such that the poles of $f$ on the real axis are precisely at the primes? I say "naturally occurring" since we can always cook up a function with the right poles of the right order, but I would hope that it comes from number theory.

Alternatively, is there a naturally arising meromorphic function whose poles appear at all primes and powers of primes?

  • $\begingroup$ What you mean is «meromorphic function with poles at the primes», not «meromorphic at the primes». $\endgroup$ – Mariano Suárez-Álvarez Feb 14 '13 at 17:14
  • $\begingroup$ no there isnt... $\endgroup$ – user58512 Feb 14 '13 at 17:38
  • $\begingroup$ Fixed the title $\endgroup$ – mck Feb 14 '13 at 17:53
  • $\begingroup$ Can you construct a function with poles at the integers? $\endgroup$ – Thom Tyrrell Feb 14 '13 at 18:25
  • 2
    $\begingroup$ Yes, the Weierstrass $\mathfrak{p}$ function for the integer lattice will have poles at exactly the integers. The poles will be order 2, but at least for now I won't put any restrictions on that. $\endgroup$ – mck Feb 14 '13 at 18:29

Let $P(s) = \sum_{p} p^{-s}$ be the Prime zeta function.

Claim: the series expansion $$f(z) = 2 \sum_{n=1}^\infty P(2n)z^{2n-1}$$ defines a meromorphic function whose only poles are simple poles of residue $1$ at the primes and their negatives.

Let $F(z) = \prod_ p (1-z^2/p^2)^{-1}$. The product converges uniformly on compact subsets of $\mathbf C - P$, where $P$ is the set of all primes and their negatives. Therefore it is holomorphic there. It has simple poles at the points of $P$. (Note the special value $F(1) = \zeta(2) = \pi^2/6$).

Using the series expansion for $\log(1-x)$, we have the series expansion for $\log F$, $$\log F(z) = \sum_{n=1}^\infty \frac{P(2n)}{n}z^{2n}.$$

Taking the derivative we get $f(z)$. Since the poles of $F$ are simple and $F$ is never $0$ on $\mathbf C - P$ (on account of the convergence of the product), $f = F'/F$ has simple poles with residue $1$ at the points of $P$, and no other poles.

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  • $\begingroup$ And this function is "naturally occuring"? $\endgroup$ – mrf Sep 27 '13 at 6:23
  • $\begingroup$ @mrf Up to a unit of the ring of holomorphic functions on $\mathbf C$, it is unique! :) $\endgroup$ – Bruno Joyal Sep 27 '13 at 6:27

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