Uniform bounds for lattice point counting and partial sums of zeta functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866909704016363520 |
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| author | Lowry-Duda, David Taniguchi, Takashi Thorne, Frank |
| author_facet | Lowry-Duda, David Taniguchi, Takashi Thorne, Frank |
| contents | 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 $Λ\subseteq \mathbb{R}^d$ 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 $ϕ(s) = \sum_n a(n) n^{-s}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_02190 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Uniform bounds for lattice point counting and partial sums of zeta functions Lowry-Duda, David Taniguchi, Takashi Thorne, Frank Number Theory 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 $Λ\subseteq \mathbb{R}^d$ 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 $ϕ(s) = \sum_n a(n) n^{-s}$ 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. |
| title | Uniform bounds for lattice point counting and partial sums of zeta functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/1710.02190 |