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The divisor function on residue classes I

Volume 168 / 2015

Prapanpong Pongsriiam, Robert C. Vaughan Acta Arithmetica 168 (2015), 369-381 MSC: Primary 11N37; Secondary 11A25, 11B25. DOI: 10.4064/aa168-4-3

Abstract

Let $d(n)$ be the number of positive divisors of $n$, and let $c_r(a)$ be Ramanujan's sum. We prove that for $q\geq 1$, $a\in \mathbb Z$, and $x\geq 1$, $$ \sum_{\substack{n\leq x\\ n\equiv a\,{\rm mod}\, q}}d(n) = \frac{x}{q} \sum_{r|q} \frac{c_r(a)}{r} \biggl({\log\frac{x}{r^2}} +2\gamma -1 \biggr) +O( (x^{1/3}+q^{1/2})x^{\varepsilon}). $$

Authors

  • Prapanpong PongsriiamDepartment of Mathematics
    Faculty of Science
    Silpakorn University
    Nakhon Pathom, 73000, Thailand
    e-mail
    e-mail
  • Robert C. VaughanDepartment of Mathematics
    McAllister Building
    Pennsylvania State University
    University Park, PA 16802-6401, U.S.A.
    e-mail

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