The divisor function on residue classes III

Prapanpong Pongsriiam, Robert C. Vaughan

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2 Scopus citations


Let n, q, a∈ N, d(n) the number of positive divisors of n, and A(x, q, a) the sum of d(n) over n≤ x and n≡a(modq). Previously we obtained an asymptotic formula for the second moment V(x,Q)=∑q≤Q∑a=1q|A(x,q,a)-M(x,q,a)|2,where Q≤ x and M(x, q, a) is a usual expected value for A(x, q, a). In this article, we compute the second moment similar to the previous one but M(x, q, a) is replaced by a better approximating function ρR(x, q, a) given by ρR(x,q,a)=∑n≤xn≡a(modq)∑r≤Rcr(n)r(lognr2+2γ),where cr(n) is Ramanujan’s sum. The advantage of using ρR(x, q, a) instead of M(x, q, a) is that the new second moment, for a suitable R, gives us practically the same main term and the error term of smaller or the same order of magnitude with much simpler and shorter calculation than that of V(x, Q).

Original languageEnglish (US)
Pages (from-to)697-719
Number of pages23
JournalRamanujan Journal
Issue number2
StatePublished - Nov 2021

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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