TY - JOUR

T1 - The divisor function on residue classes III

AU - Pongsriiam, Prapanpong

AU - Vaughan, Robert C.

N1 - Funding Information:
The first author received financial support jointly from the Thailand Research Fund and Faculty of Science, Silpakorn University, grant number RSA5980040. The second author is supported in part by NSA grant number H98230-12-1-0276.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/11

Y1 - 2021/11

N2 - 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).

AB - 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).

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U2 - 10.1007/s11139-020-00288-5

DO - 10.1007/s11139-020-00288-5

M3 - Article

AN - SCOPUS:85088581285

VL - 56

SP - 697

EP - 719

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 2

ER -