## Functiones et Approximatio Commentarii Mathematici

### A footnote to a theorem of Halász

#### 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

https://projecteuclid.org/euclid.facm/1576292536

Digital Object Identifier
doi:10.7169/facm/1847

#### 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

#### References

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