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On the Selberg class of Dirichlet series: Small degrees. (English) Zbl 0796.11037
Atle Selberg introduced a class of Dirichlet series $F\left(s\right)={\sum }_{n=1}^{\infty }{a}_{n}{n}^{-s}$ satisfying the following conditions: (1) ${\left(s-1\right)}^{m}F\left(s\right)$ is an entire function of finite order for some nonnegative integer $m$, (2) ${a}_{n}\ll {n}^{\epsilon }$, (3) a functional equation of the Riemann type holds involving a “gamma factor” of the type ${\prod }_{i=1}^{k}{\Gamma }\left({w}_{i}s+{\mu }_{i}\right)$ with ${w}_{i}>0$ and $\text{Re}\phantom{\rule{4.pt}{0ex}}{\mu }_{i}\ge 0$, (4) $F\left(s\right)$ has an Euler product. It is reasonable to conjecture that Riemann’s hypothesis holds for this class of functions. In addition, Selberg made some other conjectures whose consequences are discussed in the present paper. However, the main topic is the quantity ${d}_{F}=2{\sum }_{i=1}^{k}{w}_{i}$ called the degree of $F$. For instance, ${d}_{\zeta }=1$ for Riemann’s zeta-function. It is shown that the degree 1 is actually minimal if the constant function $F=1$ is excluded. It follows that any function in the Selberg class can be factored into a product of “primitive” functions; these are functions which cannot be written as a product of nontrivial functions of the same class. Also, $F$ is primitive if ${d}_{F}<2$. The last two sections of the paper are devoted to the important case ${w}_{i}=1/2$ when the degree is 1 or 2; in all known cases, the gamma factor can be written so that indeed ${w}_{i}=1/2$.
Reviewer: M.Jutila (Turku)

##### MSC:
 11M41 Other Dirichlet series and zeta functions
##### Keywords:
Dirichlet series; Riemann’s hypothesis; Selberg class