The divisor function on residue classes I
Volume 168 / 2015
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 $x\geq 1$, $$ \sum_{\substack{n\leq x\\ n\equiv a\,{\rm mod}\, q}}d(n) = \frac{x}{q} \sum_{r|q} \frac{c_r(a)}{r} \biggl({\log\frac{x}{r^2}} +2\gamma -1 \biggr) +O( (x^{1/3}+q^{1/2})x^{\varepsilon}). $$
