Uniform bounds for lattice point counting and partial sums of zeta functions

Abstract

We prove uniform versions of two classical results in analytic number theory. The first is an asymptotic for the number of points of a complete lattice \(\Lambda \subseteq \mathbb {R}^d\) inside the d-sphere of radius R. In contrast to previous works, we obtain error terms with implied constants depending only on d. Secondly, let \(\phi (s) = \sum _n a(n) n^{-s}\) be a ‘well behaved’ zeta function. A classical method of Landau yields asymptotics for the partial sums \(\sum _{n < X} a(n)\), with power saving error terms. Following an exposition due to Chandrasekharan and Narasimhan, we obtain a version where the implied constants in the error term will depend only on the ‘shape of the functional equation’, implying uniform results for families of zeta functions with the same functional equation.