A footnote to a theorem of Hal\'{a}sz

TL;DR

This paper investigates the zeros and poles of Dirichlet series associated with multiplicative functions bounded by 1, providing optimal bounds on their summatory functions when such zeros occur, extending Halász's theorem.

Contribution

It establishes a precise estimate for the summatory function of multiplicative functions with a zero on the critical line, refining Halász's theorem and characterizing the behavior of their Dirichlet series.

Findings

If the Dirichlet series has a zero on the one-line, the summatory function is bounded by (x / log x) times an exponential of sqrt(log log x).

The derived bounds are shown to be optimal.

The work extends Halász's theorem to include cases with zeros on the critical line.

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 Hal\'{a}sz, 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.