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 \(F(s) = \sum a_ nn^{-s}\) be a Dirichlet series with \(a_ n \ll n^ \varepsilon\) for any \(\varepsilon > 0\). 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 \(\log F(s)\) also has a Dirichlet series \(\sum b_ nn^{-s}\) with \(b_ n\) supported on the prime powers, and satisfying \(b_ n \ll n^ \vartheta\) for some \(\vartheta < {1\over 2}\). 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 \(F_ 1(s)\) and \(F_ 2(s)\) cannot be factorized into other functions of the same type then \[ \sum_{p\leq x} a_{1p}\overline{a_{2p}}/p = \delta\log\log x + O(1), \] where \(\delta = 1\) or 0 depending on whether \(F_ 1 = F_ 2\) or not.

Subject to certain hypotheses on the zeros of \(F(s)\), the value distribution of \(\log F(\sigma + it)\) for fixed \(\sigma\) near \(1\over 2\) is found, which permits an investigation of the “\(a\)-points” of \(F(s)\) (i.e. the zeros of \(F(s)-a\)). Finally similar questions for linear combinations \(\sum^ n_ 1 c_ iF_ i(s)\) are considered.

For the entire collection see [Zbl 0772.00021].

For example it is conjectured that if \(F_ 1(s)\) and \(F_ 2(s)\) cannot be factorized into other functions of the same type then \[ \sum_{p\leq x} a_{1p}\overline{a_{2p}}/p = \delta\log\log x + O(1), \] where \(\delta = 1\) or 0 depending on whether \(F_ 1 = F_ 2\) or not.

Subject to certain hypotheses on the zeros of \(F(s)\), the value distribution of \(\log F(\sigma + it)\) for fixed \(\sigma\) near \(1\over 2\) is found, which permits an investigation of the “\(a\)-points” of \(F(s)\) (i.e. the zeros of \(F(s)-a\)). Finally similar questions for linear combinations \(\sum^ n_ 1 c_ iF_ i(s)\) are considered.

For the entire collection see [Zbl 0772.00021].

Reviewer: D.R.Heath-Brown (Oxford)

##### MSC:

11M41 | Other Dirichlet series and zeta functions |

11R39 | Langlands-Weil conjectures, nonabelian class field theory |