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September 2020 A footnote to a theorem of Halász
Éric Saïas, Kristian Seip
Funct. Approx. Comment. Math. 63(1): 125-131 (September 2020). DOI: 10.7169/facm/1847

Abstract

We study multiplicative functions $f$ satisfying $|f(n)|\le 1$ for all $n$, the associated Dirichlet series $F(s):=\sum_{n=1}^{\infty} f(n) n^{-s}$, and the summatory function $S_f(x):=\sum_{n\le x} f(n)$. Up to a possible trivial contribution from the numbers $f(2^k)$, $F(s)$ may have at most one zero or one pole on the one-line, in a sense made precise by Hal\'{a}sz. We estimate $\log F(s)$ away from any such point and show that if $F(s)$ has a zero on the one-line in the sense of Halász, then $|S_f(x)|\le (x/\log x) \exp\big(c\sqrt{\log \log x}\big)$ for all $c>0$ when $x$ is large enough. This bound is best possible.

Citation

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Éric Saïas. Kristian Seip. "A footnote to a theorem of Halász." Funct. Approx. Comment. Math. 63 (1) 125 - 131, September 2020. https://doi.org/10.7169/facm/1847

Information

Published: September 2020
First available in Project Euclid: 14 December 2019

MathSciNet: MR4149514
Digital Object Identifier: 10.7169/facm/1847

Subjects:
Primary: 11M41
Secondary: 11N64

Rights: Copyright © 2020 Adam Mickiewicz University

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Vol.63 • No. 1 • September 2020
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