On Riesz Means of the Coefficients of Epstein’s Zeta Functions

Abstract

Let r_k(n) denote the number of lattice points on a k-dimensional sphere of radius \( \sqrt{n} \). The generating function

\( {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, \)

is Epstein’s zeta function. The paper considers the Riesz mean of the type

\( {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), \)

where ρ > 0; the error term Δ_ρ(x; ζ₃) is defined by

\( {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). \)

K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that

\( {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} \)

In the present paper, it is proved that

\( {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2<\rho <1\right),\\ {}O\left({x}^{3/4+\rho /4+\varepsilon}\right)& \left(0<\rho \le 1/2\right),\end{array}} \)

and the Riesz means of the coefficients of ζ_k(s), k ≥ 4, are studied.