*(English)*Zbl 0787.11037

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

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 $logF(\sigma +it)$ 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 |