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On the variance of sums of divisor functions in short intervals


Author: Stephen Lester
Journal: Proc. Amer. Math. Soc. 144 (2016), 5015-5027
MSC (2010): Primary 11N37, 11M06
DOI: https://doi.org/10.1090/proc/12914
Published electronically: August 17, 2016
MathSciNet review: 3556248
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Abstract: Given a positive integer $ n$ the $ k$-fold divisor function $ d_k(n)$ equals the number of ordered $ k$-tuples of positive integers whose product equals $ n$. In this article we study the variance of sums of $ d_k(n)$ in short intervals and establish asymptotic formulas for the variance of sums of $ d_k(n)$ in short intervals of certain lengths for $ k=3$ and for $ k \ge 4$ under the assumption of the Lindelöf hypothesis.


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Additional Information

Stephen Lester
Affiliation: School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Address at time of publication: Department of Mathematics, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden
Email: sjlester@kth.se

DOI: https://doi.org/10.1090/proc/12914
Received by editor(s): February 4, 2015
Published electronically: August 17, 2016
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n$^{o}$ 320755.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2016 American Mathematical Society

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