EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE

被引：12

作者：

Blomer, Valentin ^{[1
]}

机构：

[1] Math Inst, Bunsenstr 3-5, D-37073 Gottingen, Germany

来源：

JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU | 2020年 / 19卷 / 02期

基金：

美国国家科学基金会;

关键词：

Epstein zeta-function; quadratic forms; subconvexity; sup-norms; Sarnak's purity conjecture; multiple Dirichlet series; EIGENFUNCTIONS; VALUES; INTEGERS; SERIES; SUMS;

D O I：

10.1017/S1474748018000142

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of k -ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on GL.k/ associated with the maximal parabolic of type.k 1; 1/, and the exact sup-norm exponent is determined to be.k 2/=8 for k > 4. In particular, if k is odd, this exponent is not in 1 4 Z, which is relevant in the context of Sarnak's purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.

引用

页码：581 / 596

页数：16

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