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Proceedings of the American Mathematical Society
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... ...ac{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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-0991697-X
PII: S 0002-9939(1990)0991697-X
Article copyright: © Copyright 1990 American Mathematical Society