Functiones et Approximatio Commentarii Mathematici

A footnote to a theorem of Halász

Éric Saïas and Kristian Seip

Advance publication

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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.

Article information

Source
Funct. Approx. Comment. Math., Advance publication (2019), 7 pages.

Dates
First available in Project Euclid: 14 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.facm/1576292536

Digital Object Identifier
doi:10.7169/facm/1847

Subjects
Primary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}
Secondary: 11N64: Other results on the distribution of values or the characterization of arithmetic functions

Keywords
Halász theorem multiplicative functions prime number theorem

Citation

Saïas, Éric; Seip, Kristian. A footnote to a theorem of Halász. Funct. Approx. Comment. Math., advance publication, 14 December 2019. doi:10.7169/facm/1847. https://projecteuclid.org/euclid.facm/1576292536


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References

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