Decay of Mean Values of Multiplicative Functions

Granville, Andrew ; Soundararajan, K.

Canadian journal of mathematics, Tome 55 (2003) no. 6, pp. 1191-1230

Résumé

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 $p$ .

Détail

DOI : 10.4153/CJM-2003-047-0

Mots-clés : 11N60, 11N56, 10K20, 11N37

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