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Cramér proved the following theorem (see the announcement in [1] and [2]):

Consider the following function:

$$V(z)=\sum_k e^{\rho_kz}$$

Where $\rho_k$ runs through non trivial zeta zeros with $Im(\rho_k) > 0$

Cramér proved $V(z)$ converges for $Im(z) > 0$ and has a singularity at the origin of the type $\frac{\log(z)}{(1-e^{-z})}$ by which it means that the function

$$F(z) = 2πiV(z) -\frac{\log(z)}{(1-e^{-z})}$$

has a meromorphic continuation to all $\Bbb C$, with simple poles at the points $\pm πin$ where $n$ ranges over the integers, and at the points $\pm\log(p^m)$ where $p^m$ ranges over the prime powers.

I have following questions

  1. I'm wondering if $V(z)$ has alternate explicit expression ?
  2. (Simpler) reference where I can study about this ? ( Other than Cramér’s paper itself)

References

[1] Harald Cramér, "Sur les zéros de la fonction $\zeta(s)$" (French), Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Paris, tome 168 (Janvier-Juin), 539-541 (1919), JFM 47.0289.02.

[2] Harald Cramér, "Studien über die Nullstellen der Riemannschen Zetafunktion" (German), Math. Zeitschr. 4, 104-130 (1919), JFM 47.0289.03.

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    $\begingroup$ @DanieleTampieri thank you for an excellent edit! $\endgroup$
    – TPC
    May 19 at 17:24

1 Answer 1

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Q1: There is a functional relation for $V(z)$, but no "explicit expression" I know of.

Q2: Cramér's paper is from 1919, a modern and more extensive treatment is in On Cramér's theorem for general Euler products with functional equation (1993).

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