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
收藏
页码：581 / 596
页数：16
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