On Exponential Sums Related to the Circle Problem

  • R. Slezeviciene
  • J. Steading


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).
How to Cite
Slezeviciene, R., and J. Steading. “On Exponential Sums Related to the Circle Problem”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, no. 10, Oct. 2004, pp. 1405-18, https://umj.imath.kiev.ua/index.php/umj/article/view/3853.
Research articles