Multiplicative functions on arithmetic progressions

Hildebrand, Adolf

doi:10.1090/S0002-9939-1990-0991697-X

Multiplicative functions on arithmetic progressions
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Proc. Amer. Math. Soc. 108 (1990), 307-318 Request permission

Abstract:

Let $f$ be a multiplicative arithmetic function satisfying $\left | f \right | \leq 1$, 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 \[ \sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {n \equiv a\bmod q} \\ \end {array} } {f(n) = \frac {1}{{\varphi (q)}}} \sum \limits _{\begin {array}{*{20}{c}} {n \leq x} \\ {(n,q) = 1} \\ \end {array} } {f(n) + O\left ( {\frac {x}{q}{{\left ( {\log \frac {{\log x}}{{\log Q}}} \right )}^{ - 1/2}}} \right )} \] 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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