Geodesic
The divisor function on residue classes I
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.
Acta Arithmetica, Tome 168 (2015) no. 4, pp. 369-381 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $x\geq 1$, $$ \sum_{\substack{n\leq x\\ n\equiv a\,{\rm mod}\, q}}d(n) = \frac{x}{q} \sum_{r|q} \frac{c_r(a)}{r} \biggl({\log\frac{x}{r^2}} +2\gamma -1 \biggr) +O( (x^{1/3}+q^{1/2})x^{\varepsilon}). $$
DOI : 10.4064/aa168-4-3
Keywords: number positive divisors ramanujans sum prove geq mathbb geq sum substack leq equiv mod frac sum frac biggl log frac gamma biggr varepsilon

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. 
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     title = {The divisor function on residue classes {I}},
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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

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