Averages of shifted convolutions of general divisor sums involving Hecke eigenvalues

2021 ◽  
Author(s):  
Dan Wang
Keyword(s):  
Hecke Eigenvalues ◽  
Divisor Sums

Averages of shifted convolutions of general divisor sums involving Hecke eigenvalues

2019 ◽  
Vol 199 ◽  
pp. 342-351
Author(s):  
Guangshi Lü ◽  
Dan Wang
Keyword(s):  
Hecke Eigenvalues ◽  
Divisor Sums

scholarly journals Quantitative versions of the joint distributions of Hecke eigenvalues

2016 ◽  
Vol 169 ◽  
pp. 295-314 ◽  
Author(s):  
Hengcai Tang ◽  
Yingnan Wang
Keyword(s):  
Joint Distributions ◽  
Hecke Eigenvalues

Weighted divisor sums and Bessel function series, IV

2012 ◽  
Vol 29 (1-3) ◽  
pp. 79-102 ◽  
Author(s):  
Bruce C. Berndt ◽  
Sun Kim ◽  
Alexandru Zaharescu
Keyword(s):  
Bessel Function ◽  
Function Series ◽  
Divisor Sums

Power moments of Hecke eigenvalues for congruence group

2019 ◽  
Vol 198 ◽  
pp. 139-158 ◽  
Author(s):  
Ping Song ◽  
Wenguang Zhai ◽  
Deyu Zhang
Keyword(s):  
Congruence Group ◽  
Hecke Eigenvalues ◽  
Power Moments

scholarly journals The orthogonality of Hecke eigenvalues

2007 ◽  
Vol 143 (03) ◽  
pp. 541-565 ◽  
Author(s):  
Henryk Iwaniec ◽  
Xiaoqing Li
Keyword(s):  
Hecke Eigenvalues

Detecting Large Simple Rational Hecke Modules for Γ0(N) via Congruences

2018 ◽  
Vol 2020 (19) ◽  
pp. 6149-6168
Author(s):  
Michael Lipnowski ◽  
George J Schaeffer
Keyword(s):  
Modular Forms ◽  
Class Number ◽  
Target Space ◽  
Large Set ◽  
Hecke Eigenvalues ◽  
Artin's Conjecture ◽  
Number Problem ◽  
Novel Method ◽  
Hecke Modules ◽  
Primitive Roots

Abstract We describe a novel method for bounding the dimension $d$ of the largest simple Hecke submodule of $S_{2}(\Gamma _{0}(N);\mathbb{Q})$ from below. Such bounds are of interest because of their relevance to the structure of $J_{0}(N)$, for instance. In contrast with previous results of this kind, our bound does not rely on the equidistribution of Hecke eigenvalues. Instead, it is obtained via a Hecke-compatible congruence between the target space and a space of modular forms whose Hecke eigenvalues are easily controlled. We prove conditional bounds, the strongest of which is $d\gg _{\epsilon } N^{1/2-\epsilon }$ over a large set of primes $N$, contingent on Soundararajan’s heuristics for the class number problem and Artin’s conjecture on primitive roots. For prime levels $N\equiv 7\mod 8,$ our method yields an unconditional bound of $d\geq \log _{2}\log _{2}(\frac{N}{8})$, which is larger than the known bound of $d\gg \sqrt{\log \log N}$ due to Murty–Sinha and Royer. A stronger unconditional bound of $d\gg \log N$ can be obtained in more specialized (but infinitely many) cases. We also propose a number of Maeda-style conjectures based on our data, and we outline a possible congruence-based approach toward the conjectural Hecke simplicity of $S_{k}(\textrm{SL}_{2}(\mathbb{Z});\mathbb{Q})$.

scholarly journals Hecke eigenvalues and relations for degree 2 Siegel Eisenstein series

2012 ◽  
Vol 132 (11) ◽  
pp. 2700-2723 ◽  
Author(s):  
Lynne H. Walling
Keyword(s):  
Eisenstein Series ◽  
Hecke Eigenvalues

Divisor Sums of Generalised Exponential Polynomials

1996 ◽  
Vol 39 (1) ◽  
pp. 35-46 ◽  
Author(s):  
G. R. Everest ◽  
I. E. Shparlinski
Keyword(s):  
Prime Ideal ◽  
Algebraic Integer ◽  
Linear Recurrence ◽  
Exponential Polynomial ◽  
Exponential Polynomials ◽  
Recurrence Sequence ◽  
Special Cases ◽  
Divisor Sums ◽  
Recurrence Sequences

AbstractA study is made of sums of reciprocal norms of integral and prime ideal divisors of algebraic integer values of a generalised exponential polynomial. This includes the important special cases of linear recurrence sequences and general sums of S-units. In the case of an integral binary recurrence sequence, similar (but stronger) results were obtained by P. Erdős, P. Kiss and C. Pomerance.

Some problems involving Hecke eigenvalues

2019 ◽  
Vol 159 (1) ◽  
pp. 287-298
Author(s):  
H. F. Liu ◽  
R. Zhang
Keyword(s):  
Hecke Eigenvalues
Sign in / Sign up

Export Citation Format

Share Document