Short sums of multiplicative functions

Kapoor, Vishaal

doi:10.1090/S0002-9939-2012-11257-2

Short sums of multiplicative functions
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Proc. Amer. Math. Soc. 140 (2012), 3693-3701 Request permission

Abstract:

We show that on a short interval, $x < n \leq x+w$, the average value of a complex-valued multiplicative function $f(n)$ that is sufficiently close to $1$ on primes and bounded on prime powers, tends to \begin{align*} C_f = \prod _{p} \bigg (1-\frac 1p\bigg )\bigg (1+\frac {f(p)}p + \frac {f(p^2)}{p^2} + ...\bigg ), \end{align*} provided the interval is sufficiently long with respect to $x$.

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