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

TL;DR

This paper establishes uniform bounds for lattice point counts within spheres and for partial sums of zeta functions, with error terms depending only on dimension and shape of the functional equation, respectively.

Contribution

It provides the first uniform error bounds for lattice point counting and zeta function partial sums that depend solely on dimension and functional equation shape.

Findings

Error terms depend only on dimension for lattice point counts.

Uniform bounds for zeta function partial sums based on functional equation shape.

Results apply uniformly across families of zeta functions with the same functional equation.

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.