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A remark on divisor-weighted sums

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Abstract

Let \(\{a_n\}\) be a sequence of nonnegative real numbers. Under very mild hypotheses, we obtain upper bounds of the expected order of magnitude for sums of the form \(\sum _{n \le x} a_n \tau _r(n)\), where \(\tau _r(n)\) is the \(r\)-fold divisor function. This sharpens previous estimates of Friedlander and Iwaniec. The proof uses combinatorial ideas of Erdős and Wolke.

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Acknowledgments

Thanks are due to Enrique Treviño for carefully reading through the manuscript and suggesting a number of changes that improved the exposition. Our proof of Theorem 1.2 was influenced by a wonderfully lucid blog post of Terry Tao [10] expounding on Erdős’s paper [1].

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Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, 30602, USA

    Paul Pollack

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  1. Paul Pollack
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Correspondence to Paul Pollack.

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This work was supported by NSF award DMS-1402268.

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Pollack, P. A remark on divisor-weighted sums. Ramanujan J 40, 63–69 (2016). https://doi.org/10.1007/s11139-014-9669-1

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  • DOI: https://doi.org/10.1007/s11139-014-9669-1

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