Zeros of partial sums of \(L\)-functions. (English) Zbl 1494.11077
Summary: We consider a certain class of multiplicative functions \(f : \mathbb{N} \rightarrow \mathbb{C}\). Let \(F(s) = \sum_{n = 1}^\infty f(n) n^{- s}\) be the associated Dirichlet series and \(F_N(s) = \sum_{n \leq N} f(n) n^{- s}\) be the truncated Dirichlet series. In this setting, we obtain new Halász-type results for the logarithmic mean value of \(f\). More precisely, we prove estimates for the sum \(\sum_{n = 1}^x f(n) / n\) in terms of the size of \(| F(1 + 1 / \log x) |\) and show that these estimates are sharp. As a consequence of our mean value estimates, we establish non-trivial zero-free regions for the partial sums \(F_N(s)\).
In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field \(K\). More precisely, we give some improved results for the number of zeros up to height \(T\) as well as new zero density results for the number of zeros up to height \(T\), lying to the right of \(\operatorname{Re}(s) = \sigma\), where \(\sigma \geq 1 / 2\).
In particular, we study the zero distribution of partial sums of the Dedekind zeta function of a number field \(K\). More precisely, we give some improved results for the number of zeros up to height \(T\) as well as new zero density results for the number of zeros up to height \(T\), lying to the right of \(\operatorname{Re}(s) = \sigma\), where \(\sigma \geq 1 / 2\).
MSC:
| 11M41 | Other Dirichlet series and zeta functions |
| 11N37 | Asymptotic results on arithmetic functions |
| 11M26 | Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses |
| 11N64 | Other results on the distribution of values or the characterization of arithmetic functions |
Keywords:
mean values of multiplicative functions; zeros of exponential sums; \(L\)-functions; Dirichlet polynomials; Dedekind zeta function; distribution of zerosReferences:
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