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Decay of Mean Values of Multiplicative Functions

Published online by Cambridge University Press:  20 November 2018

Andrew Granville  and
K. Soundararajan
Andrew Granville
Affiliation:
Départment de Mathématiques, et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, Québec, H3C 3J7 e-mail: andrew@dms.umontreal.ca
K. Soundararajan
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109, USA e-mail: ksound@math.lsa.umich.edu
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Abstract

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For given multiplicative function $f$, with $\left| f\left( n \right) \right|\le 1$ for all $n$, we are interested in how fast its mean value $\left( 1/x \right)\sum{_{n\le x}f\left( n \right)}$ converges. Halász showed that this depends on the minimum $M\,\left( \text{over}\,y\,\in \,\mathbb{R} \right)\,\text{of}\,\sum{_{p\le x}\left( 1-\operatorname{Re}\left( f\left( p \right){{p}^{-iy}} \right) \right)}/p$, and subsequent authors gave the upper bound $\ll \left( 1+M \right){{e}^{-M}}$. 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 Halász-Montgomery 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 $p$.

Keywords

11N6011N5610K2011N37
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 6 , 01 December 2003 , pp. 1191 - 1230
Copyright
Copyright © Canadian Mathematical Society 2003

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