2018
Том 70
№ 6

# On Exponential Sums Related to the Circle Problem

Abstract

Let r(n) count the number of representations of a positive integer n as a sum of two integer squares. We prove a truncated Voronoi-type formula for the twisted Mobius transform $$\mathop \sum \limits_{n \leqslant x} \;\,r(n)\;\exp \left( {2\pi i\frac{{nk}}{{4l}}} \right),$$ where k and l are positive integers such that k and 4l are coprime, and give some applications (almost periodicity, limit distribution, an asymptotic mean-square formula, and O- and Ω-estimates for the error term).

English version (Springer): Ukrainian Mathematical Journal 56 (2004), no. 10, pp 1676-1692.

Citation Example: Slezeviciene R., Steading J. On Exponential Sums Related to the Circle Problem // Ukr. Mat. Zh. - 2004. - 56, № 10. - pp. 1405-1418.

Full text