Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Multiplicative functions on arithmetic progressions

Author: Adolf Hildebrand
Journal: Proc. Amer. Math. Soc. 108 (1990), 307-318
MSC: Primary 11N64; Secondary 11N37
MathSciNet review: 991697
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Abstract: Let $ f$ be a multiplicative arithmetic function satisfying $ \left\vert f \right\vert \leq 1$, let $ x \geq 10$ and $ 2 \leq Q \leq {x^{1/3}}$. It Is shown that, with suitable integers $ {q_1} \geq 2$ and $ {q_2} \geq 2$, the estimate

$\displaystyle \sum\limits_{\begin{array}{*{20}{c}} {n \leq x} \\ {n \equiv a\bm...{x}{q}{{\left( {\log \frac{{\log x}}{{\log Q}}} \right)}^{ - 1/2}}} \right)} $

holds uniformly for $ \left( {a,q} \right) = 1$ and all moduli $ q \leq Q$ that are not multiples of $ {q_1}$ or $ {q_2}$.

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Article copyright: © Copyright 1990 American Mathematical Society