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Old and new conjectures and results about a class of Dirichlet series. (English) Zbl 0787.11037
Bombieri, E. (ed.) et al., Proceedings of the Amalfi conference on analytic number theory, held at Maiori, Amalfi, Italy, from 25 to 29 September, 1989. Salerno: Universitá di Salerno, 367-385 (1992).

Let $F\left(s\right)=\sum {a}_{n}{n}^{-s}$ be a Dirichlet series with ${a}_{n}\ll {n}^{\epsilon }$ for any $\epsilon >0$. Assume that there is an analytic continuation to an entire function, except possibly for a pole at $s=1$, and suppose there is a functional equation of the usual type. Suppose further that $logF\left(s\right)$ also has a Dirichlet series $\sum {b}_{n}{n}^{-s}$ with ${b}_{n}$ supported on the prime powers, and satisfying ${b}_{n}\ll {n}^{\vartheta }$ for some $\vartheta <\frac{1}{2}$. Various conjectures on such functions are presented, which can be viewed as a very low-brow alternative to the Langlands philosophy.

For example it is conjectured that if ${F}_{1}\left(s\right)$ and ${F}_{2}\left(s\right)$ cannot be factorized into other functions of the same type then

$\sum _{p\le x}{a}_{1p}\overline{{a}_{2p}}/p=\delta loglogx+O\left(1\right),$

where $\delta =1$ or 0 depending on whether ${F}_{1}={F}_{2}$ or not.

Subject to certain hypotheses on the zeros of $F\left(s\right)$, the value distribution of $logF\left(\sigma +it\right)$ for fixed $\sigma$ near $\frac{1}{2}$ is found, which permits an investigation of the “$a$-points” of $F\left(s\right)$ (i.e. the zeros of $F\left(s\right)-a$). Finally similar questions for linear combinations ${\sum }_{1}^{n}{c}_{i}{F}_{i}\left(s\right)$ are considered.

MSC:
 11M41 Other Dirichlet series and zeta functions 11R39 Langlands-Weil conjectures, nonabelian class field theory