Proceedings of the American Mathematical Society

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



Short sums of multiplicative functions

Author: Vishaal Kapoor
Journal: Proc. Amer. Math. Soc. 140 (2012), 3693-3701
MSC (2010): Primary 11N37
Published electronically: March 2, 2012
MathSciNet review: 2944710
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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

$\displaystyle C_f = \prod _{p} \bigg (1-\frac 1p\bigg )\bigg (1+\frac {f(p)}p + \frac {f(p^2)}{p^2} + ...\bigg ),$    

provided the interval is sufficiently long with respect to $ x$.

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Vishaal Kapoor
Affiliation: Google, 1600 Amphitheatre Parkway, Mountain View, California 94043

Received by editor(s): October 6, 2010
Received by editor(s) in revised form: April 21, 2011
Published electronically: March 2, 2012
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.