Uniform bounds for lattice point counting and partial sums of zeta functions
Uniform bounds for lattice point counting and partial sums of zeta functions
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DOI：
10.1007/s00209-021-02862-z
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发表时间：
2017-10
影响因子：
0.8
通讯作者：
David Lowry-Duda;Takashi Taniguchi;F. Thorne
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文献类型：
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作者：
David Lowry-Duda;Takashi Taniguchi;F. Thorne
本文证明了解析数论中两个经典结果的统一形式。第一个是半径为R的d-球面内完备格的点数的渐近性。与以前的作品相比，我们得到的误差项与隐含的常数只取决于OND。第二，让我们成为一个“行为良好”的zeta函数。一个经典的方法朗道产量的部分和渐近，与节能的误差项。以下的阐述由于rasekharan和Narasimhan，我们得到一个版本，其中隐含的常数在误差项将只取决于“形状的功能方程”，这意味着统一的结果为家庭的zeta函数具有相同的功能方程。
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 latticeinside thed-sphere of radiusR. In contrast to previous works, we obtain error terms with implied constants depending only ond. Secondly, letbe a ‘well behaved’ zeta function. A classical method of Landau yields asymptotics for the partial sums, 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.