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I am trying to find an estimate for the following sum:

$$ \sum_{\substack{n \leq x \\ n \equiv k (m)}} d(n), $$

where $d(n)$ is number of divisors of $n$. I found estimates for the case when $k$ and $m$ are coprime, but nothing explicit for general case. My expectation for this sum is that this should be known, but I can't find anything. Any ideas/references?

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2 Answers 2

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In the context of the "divisor problem for arithmetical progressions", there is the following article by Fouvry, Iwaniec and Katz, on "The divisor function over arithmetic progressions": http://matwbn.icm.edu.pl/ksiazki/aa/aa61/aa6134.pdf. The authors use Fourier series techniques and Weil’s estimate for Kloosterman sums to obtain $$ S(x,m,k)-D(x,m)=\sum_{n\le x, n\equiv k(m)}d(n)-\frac{1}{\phi(m)}\sum_{n\le x,(n,m)=1}d(n)=O((m^{1/2}+x^{1/3})x^{\epsilon}). $$ We have the estimate $$ \phi(m)D(x,m)=\sum_{n\le x,(n,m)=1}d(n)=xP(\log x)+O(x^{1/2}m^{\epsilon}), $$ with $P(\log x)=\frac{\phi(m)^2}{m^2}(\log x+\gamma-1)+2\frac{\phi(m)}{m}\sum_{d\mid m}\frac{\mu(d)\log d}{d}$.

Werner Georg Nowak proved in $1984$ that for any given natural numbers $m$ and $k$, $$ S(x,m,k)=\alpha x(\log x+2\gamma -1)+\beta x+ O(x^{35/108+\epsilon}), $$ for constants $\alpha$ and $\beta$ depending only on $m$ and $k$.

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    $\begingroup$ Are you sure that you have stated the upper bounds correctly? Already for $m=k=1$, it is known that the sum is asymptotic to $x\log x$ as $x \to \infty$. Or are you referring to the error terms in the corresponding Dirichlet divisor problem? $\endgroup$
    – Daniel Loughran
    Commented Nov 18, 2013 at 17:23
  • $\begingroup$ @DanielLoughran Yes, the difference, sorry. I edited my answer. $\endgroup$
    – Dietrich Burde
    Commented Nov 18, 2013 at 19:06
  • $\begingroup$ Those authors consider the situation where $m$ and $k$ are coprime. $\endgroup$
    – Matt Young
    Commented Nov 18, 2013 at 19:39
  • $\begingroup$ Thank you for the answer, but this is indeed, as Matt noted, the coprime case $\endgroup$
    – psarka
    Commented Nov 19, 2013 at 6:37
  • $\begingroup$ @PauliusŠarka I don't think so. Nowak's estimates are for all natural numbers $m,k$. $\endgroup$
    – Dietrich Burde
    Commented Nov 19, 2013 at 8:58
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Werner Georg Nowak, ON A RESULT OF SMITH AND SUBBARAO CONCERNING A DIVISOR PROBLEM Canad. Math. Bull. Vol. 27 (4), 1984.

Werner considers a different problem. He investigates the number of (positive) divisors of the positive integer n which are congruent to l modulo k. I.e. d(n;l,k) = #{d : d|n & d ≡ l(k)}.

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  • $\begingroup$ No, I mean W. G. Nowak, On the divisor problem in arithmetic progressions, Comment. Math. Univ. St. Pauli 33 (1984), 209–217. $\endgroup$
    – Dietrich Burde
    Commented Nov 19, 2013 at 9:10
  • $\begingroup$ where is this paper? i couldn't find it, not even on the author's website. all the results i find from him are to do with the quantity $d(n;l,k)$ in this answer and not with $S(x,m,k)$ of your answer. maybe i'm overlooking something? :) $\endgroup$
    – tomos
    Commented Jan 14, 2022 at 10:55
  • $\begingroup$ Nowak's paper is available here: sciencedirect.com/science/article/pii/0022314X89900188 $\endgroup$
    – Ramin
    Commented Dec 6 at 0:15

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