# On Cramér's theorem about roots of Zeta function

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 ?

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.

• @DanieleTampieri thank you for an excellent edit!
– TPC
May 19 at 17:24

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