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.
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Acknowledgements
We would like to thank Bruce Berndt, Eric Gaudron, Aleksandar Ivić, Jesse Kass, Martin Nowak, Anders Södergren, Jesse Thorner, Martin Widmer, and an anonymous referee for their advice, suggestions, and encouragement. We began this paper at the Mathematical Sciences Research Insitute in Berkeley, CA in Spring 2017, and we would like to thank MSRI for providing an excellent atmosphere in which to work, as well as the National Science Foundation (under Grant No. DMS-1440140) for their financial support of MSRI. The first author was partially supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 0228243, the EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms, and the Simons Collaboration in Arithmetic Geometry, Number Theory, and Computation via the Simons Foundation grant 546235; the second author was partially supported by the JSPS, KAKENHI Grant Numbers JP24654005, JP25707002, and JP17H02835; the third author was partially supported by the National Security Agency under Grant No. H98230-16-1-0051 and by the Simons Foundation under Grant No. 586594.
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Lowry-Duda, D., Taniguchi, T. & Thorne, F. Uniform bounds for lattice point counting and partial sums of zeta functions. Math. Z. 300, 2571–2590 (2022). https://doi.org/10.1007/s00209-021-02862-z
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DOI: https://doi.org/10.1007/s00209-021-02862-z