1Department of Mathematics Faculty of Science Silpakorn University Nakhon Pathom, 73000, Thailand 2Department of Mathematics McAllister Building Pennsylvania State University University Park, PA 16802-6401, U.S.A.
Acta Arithmetica, Tome 168 (2015) no. 4, pp. 369-381
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 $x\geq 1$,
$$
\sum_{\substack{n\leq x\\ n\equiv a\,{\rm mod}\, q}}d(n) = \frac{x}{q} \sum_{r|q} \frac{c_r(a)}{r} \biggl({\log\frac{x}{r^2}} +2\gamma -1 \biggr) +O( (x^{1/3}+q^{1/2})x^{\varepsilon}).
$$
Keywords:
number positive divisors ramanujans sum prove geq mathbb geq sum substack leq equiv mod frac sum frac biggl log frac gamma biggr varepsilon
Affiliations des auteurs :
Prapanpong Pongsriiam
1
;
Robert C. Vaughan
2
1
Department of Mathematics Faculty of Science Silpakorn University Nakhon Pathom, 73000, Thailand
2
Department of Mathematics McAllister Building Pennsylvania State University University Park, PA 16802-6401, U.S.A.
@article{10_4064_aa168_4_3,
author = {Prapanpong Pongsriiam and Robert C. Vaughan},
title = {The divisor function on residue classes {I}},
journal = {Acta Arithmetica},
pages = {369--381},
year = {2015},
volume = {168},
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doi = {10.4064/aa168-4-3},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/aa168-4-3/}
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Prapanpong Pongsriiam; Robert C. Vaughan. The divisor function on residue classes I. Acta Arithmetica, Tome 168 (2015) no. 4, pp. 369-381. doi: 10.4064/aa168-4-3
