Functiones et Approximatio Commentarii Mathematici

A footnote to a theorem of Halász

Éric Saïas and Kristian Seip

Advance publication

This article is in its final form and can be cited using the date of online publication and the DOI.

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 14 December 2019

Permanent link to this document

Digital Object Identifier

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

Halász theorem multiplicative functions prime number theorem


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.

Export citation


  • [1] A. Granville, A. Harper, and K. Soundararajan, A more intuitive proof of a sharp version of Halász's theorem, Proc. Amer. Math. Soc. 146 (2018), 4099–4104.
  • [2] A. Granville and K. Soundararajan, Decay of mean values of multiplicative functions, Canad. J. Math. 55 (2003), 1191–1230.
  • [3] G. Halász, Über die Mittelwerte multiplikativer zahlentheoretischer Funktionen, Acta Math. Acad. Sci. Hungar. 19 (1968), 365–403.
  • [4] G. Halász, On the distribution of additive and mean-values of multiplicative functions, Studia Sci. Math. Hungar. 6 (1971), 211–233.
  • [5] H.L. Montgomery, A note on mean values of multiplicative functions, Institut Mittag-Leffler Report 17, 1978.
  • [6] H.L. Montgomery, personal communication, 2015.
  • [7] K. Seip, Universality and distribution of zeros and poles of some zeta functions, to appear in J. Anal. Math.; arXiv:1812.11729.
  • [8] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd ed., Graduate Studies in Mathematics 163, American Mathematical Society, Providence, RI, 2015.