On Riesz means of the coefficients of Epstein's zeta functions

O. M. Fomenko

Zapiski Nauchnykh Seminarov POMI, Analytical theory of numbers and theory of functions. Part 33, Tome 458 (2017), pp. 218-235

Résumé

Let $r_k(n)$ denote the number of lattice points on a $k$-dimensional sphere of radius $\sqrt n$.The generating function $$ \zeta_k(s)=\sum^\infty_{n=1}r_k(n)n^{-s},\ k\geq2, $$ is Epstein's zeta-function. Let $k=3$. We consider the Riesz mean of the type $$ D_\rho(x;\zeta_3)=\frac1{\Gamma(\rho+1)}\sum_{n\leq x}(x-n)^\rho r_3(n) $$ for any fixed $\rho>0$ and define the error term $\Delta_\rho(x;\zeta_3)$ by $$ D_\rho(x;\zeta_3)=\frac{\pi^{3/2}x^{\rho+3/2}}{\Gamma(\rho+5/2)}+\frac{x^\rho}{\Gamma(\rho+1)}\zeta_3(0)+\Delta_\rho(x;\zeta_3). $$ A result of K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) gives $$ \Delta_\rho(x;\zeta_3)= \begin{cases} O(x^{1/2+\rho/2)}(\rho>1),\\ \Omega_\pm(x^{1/2+\rho/2})(\rho\geq0). \end{cases} $$ In § 2 one proves that $$ \Delta_\rho(x;\zeta_3)= \begin{cases} O(x\log x)(\rho=1),\\ O(x^{2/3+\rho/3+\epsilon})(1/2\rho1),\\ O(x^{3/4+\rho/4+\epsilon})(0\rho\leq1/2). \end{cases} $$ In § 3 one mentions a few examples for which results of § 2 are applicable. In § 4 one investigates Riesz means of the coefficients of $\zeta_k(s)$, $k\geq4$.