Abstract view
Decay of Mean Values of Multiplicative Functions


Published:20031201
Printed: Dec 2003
Andrew Granville
K. Soundararajan
Abstract
For given multiplicative function $f$, with $f(n) \leq 1$ for all
$n$, we are interested in how fast its mean value $(1/x) \sum_{n\leq
x} f(n)$ converges. Hal\'asz showed that this depends on the minimum
$M$ (over $y\in \mathbb{R}$) of $\sum_{p\leq x} \bigl( 1  \Re (f(p)
p^{iy}) \bigr) / p$, and subsequent authors gave the upper bound $\ll
(1+M) e^{M}$. For many applications it is necessary to have explicit
constants in this and various related bounds, and we provide these via
our own variant of the Hal\'aszMontgomery lemma (in fact the constant
we give is best possible up to a factor of 10). We also develop a new
type of hybrid bound in terms of the location of the absolute value of
$y$ that minimizes the sum above. As one application we give bounds
for the least representatives of the cosets of the $k$th powers
mod~$p$.