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MULTIPLICATIVE FUNCTIONS ON ARITHMETIC PROGRESSIONS. VII: LARGE MODULI

Published online by Cambridge University Press:  24 March 2003

P. D. T. A. ELLIOTT
Affiliation:
Department of Mathematics, University of Colorado, Campus Box 395, Boulder, CO 80309-0395, USApdtae@euclid.colorado.edu
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Abstract

A complex valued function $g$ , defined on the positive integers, is multiplicative if it satisfies $g(ab) = g(a)g(b)$ whenever the integers $a$ and $b$ are mutually prime.

THEOREM 1. Let $D$ be an integer, $2 \le D \le x, \varepsilon > 0$ . Let $g$ be a multiplicative function with values in the complex unit disc.

There is a character $\chi_1({\rm mod}\, D)$ , real if $g$ is real, such that when $0 < \gamma < 1$ , \[ \sum_{\overset{n \le y}{n\equiv a({\rm mod}\, D)}} g(n)- \frac{1}{\phi(D)} \sum_{\overset{n\le y}{(n,D)=1)}} g(n)-\frac{\overline{\chi_1(a)}}{\phi(D)} \sum_{n\le y} g(n)\chi_1(n) \ll \frac{y}{\phi(D)} \left(\frac{\log D}{\log y}\right)^{1/4-\varepsilon} \] uniformly for $(a, D) = 1, D \le y, x^\gamma \le y \le x$ , the implied constant depending at most upon $\varepsilon, \gamma$ .

Type
Notes and Papers
Copyright
The London Mathematical Society, 2002

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