Kloosterman Sums and Maass Cusp Forms of Half Integral Weight for the Modular Group

被引:12
作者
Ahlgren, Scott [1 ]
Andersen, Nickolas [2 ]
机构
[1] Univ Illinois, Dept Math, 1409 W Green St, Urbana, IL 61801 USA
[2] Univ Calif Los Angeles, Dept Math, Los Angeles, CA 90095 USA
关键词
BESSEL-FUNCTION; SERIES;
D O I
10.1093/imrn/rnw234
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We estimate the sums Sigma(c <= X) S(m, n, c, chi)/c , where the S(m, n, c, chi) are Kloosterman sums associated with a multiplier system chi of half-integral weight on the modular group. Our estimates are uniform in m, n, and x in analogy with Sarnak and Tsimerman's improvement of Kuznetsov's bound for the ordinary Kloosterman sums. Among other things this requires us to develop mean value estimates for coefficients of Maass cusp forms of weight 1/2 and uniform estimates for K-Bessel integral transforms. As an application, we obtain an improved estimate for the classical problem of estimating the size of the error term in Rademacher's formula for the partition function p(n).
引用
收藏
页码:492 / 570
页数:79
相关论文
共 52 条
[31]  
Linnik J.V., 1963, Proc. Int. Cong. Math., V130, P270
[32]  
LMFDB Collaboration, 2013, L FUNCT MOD FORMS DA
[33]  
Maaβ H., 1949, MATH ANN, V121, P141, DOI DOI 10.1007/BF01329622
[34]  
Maaβ H., 1952, MATH ANN, V125, P235, DOI DOI 10.1007/BF01343120
[35]   MODULAR FORMS OF HALF INTEGRAL WEIGHT AND INTEGRAL OF CERTAIN THETA-FUNCTIONS [J].
NIWA, S .
NAGOYA MATHEMATICAL JOURNAL, 1975, 56 (JAN) :147-161
[36]   A generalization of the Goldfeld-Sarnak estimate on Selberg's Kloosterman zeta-function [J].
Pribitkin, WD .
FORUM MATHEMATICUM, 2000, 12 (04) :449-459
[37]  
Proskurin N. V., 1979, RUSSIAN MATH J, V82, P103
[38]  
Proskurin N. V., 2003, ZAP NAUCHN SEM S PET, V302, P107
[39]  
Rademacher H, 1937, P LOND MATH SOC, V43, P241
[40]   On the expansion of the partition function in a series [J].
Rademacher, H .
ANNALS OF MATHEMATICS, 1943, 44 :416-422