Short sums of multiplicative functions
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- by Vishaal Kapoor
- Proc. Amer. Math. Soc. 140 (2012), 3693-3701
- DOI: https://doi.org/10.1090/S0002-9939-2012-11257-2
- Published electronically: March 2, 2012
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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 \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 $x$.References
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Bibliographic Information
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3693-3701
- MSC (2010): Primary 11N37
- DOI: https://doi.org/10.1090/S0002-9939-2012-11257-2
- MathSciNet review: 2944710