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Zero-free regions for Dirichlet series. (English) Zbl 1322.11091
The authors study zero-free regions of general Dirichlet series (\(L\)-functions) of the form \[ L(s) = \sum_{n=1}^\infty a_n n^{-s}, \] where some mild conditions, such as \(a_n \ll_\varepsilon n^\varepsilon\), are assumed. They study a general Beurling-Nyman criterion for the zeros of \(L\)-functions. The well-known criterion of B. Nyman [On some groups and semi-groups of translations. (PhD Thesis) (1950)] states that the Riemann hypothesis (all complex zeros of the Riemann zeta-function \(\zeta(s)\) have real parts 1/2) is equivalent to the statement that the characteristic function of the interval (0,1) belongs to the closure of \(\mathcal N\) in \(L^2(0,1)\), the set of functions \[ f(x) = \sum_{j=1}^n c_j\left\{\frac{\theta_j}{x}\right\}\quad(0 < \theta_j \leq 1, c_j \in \mathbb C, \,{y} = y - [y]), \] and \(\sum_{j=1}^nc_j\theta_j =0\).
In their investigations the authors primarily use methods from functional analysis, and they are also interested in sharpening and generalizing the work of N. Nikolski [Ann. Inst. Fourier 45, No. 1, 143–159 (1995; Zbl 0816.30026)] and A. de Roton [Acta Arith. 126, No. 1, 27–55 (2007; Zbl 1131.11062)]. In particular, the authors generalize Nikolski’s work to the well-known class \(\mathcal S\) of the Selberg class of \(L\)-functions. In section 2 four theorems of quite a general nature are presented, and since their formulations are technical, they will not be reproduced here. Applications to the zero-free regions for, \(L(s,\chi)\) and functions of \(\mathcal S\) are given in section 7.

MSC:
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
30H10 Hardy spaces
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