A footnote to a theorem of Halász

Abstract

A BSTRACT . We study multiplicative functions f satisfying | f ( n ) | ≤ 1 for all n , the associated Dirichlet series F ( s ) : = P ∞ n = 1 f ( n ) n − s , and the summatory function S f ( x ) : = P n ≤ 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 Halász. 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 ) | ≤ ( x /log x )exp ¡ c p loglog x ¢ for all c > 0 when x is large enough. This bound is best possible.