TY - JOUR

T1 - On the average sum of the kth divisor function over values of quadratic polynomials

AU - Lapkova, Kostadinka

AU - Zhou, Nian Hong

N1 - Funding Information:
Kostadinka Lapkova is supported by a Hertha Firnberg grant [T846-N35] of the Austrian Science Fund (FWF). Nian Hong Zhou is supported by the National Science Foundation of China (Grant No. 11971173).
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2021/8

Y1 - 2021/8

N2 - Let F(x) ∈ Z[x1, x2, … , xn] , n≥ 3 , be an n-variable quadratic polynomial with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum Σk,F(X;B)=∑x∈XB∩Znτk(F(x));for X tending to infinity, where B⊂ Rn is an n-dimensional box such that min x∈XBF(x) ≥ 0 for all sufficiently large X, and τk(·) is the kth divisor function for any integer k≥ 2.

AB - Let F(x) ∈ Z[x1, x2, … , xn] , n≥ 3 , be an n-variable quadratic polynomial with a nonsingular quadratic part. Using the circle method we derive an asymptotic formula for the sum Σk,F(X;B)=∑x∈XB∩Znτk(F(x));for X tending to infinity, where B⊂ Rn is an n-dimensional box such that min x∈XBF(x) ≥ 0 for all sufficiently large X, and τk(·) is the kth divisor function for any integer k≥ 2.

KW - Circle method

KW - Divisor functions

KW - Quadratic polynomials

UR - http://www.scopus.com/inward/record.url?scp=85110057525&partnerID=8YFLogxK

U2 - 10.1007/s11139-019-00240-2

DO - 10.1007/s11139-019-00240-2

M3 - Article

VL - 55

SP - 849

EP - 872

JO - The Ramanujan Journal

JF - The Ramanujan Journal

SN - 1382-4090

IS - 3

ER -