A footnote to a theorem Halász
Journal article, Peer reviewed
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Date
2020Metadata
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Original version
Functiones et Approximatio Commentarii Mathematici. 2020, 63 125-131. 10.7169/facm/1847Abstract
We study multiplicative functions f satisfying |f(n)|≤1 for all n, the associated Dirichlet series F(s):=∑∞n=1f(n)n−s, and the summatory function Sf(x):=∑n≤xf(n). Up to a possible trivial contribution from the numbers f(2k), 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 logF(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 |Sf(x)|≤(x/logx)exp(cloglogx−−−−−−−√) for all c>0 when x is large enough. This bound is best possible.