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On sums of Fourier coefficients of cusp forms
TLDR
In this paper, the summatory function of non-holomorphic cusp forms is estimated, where the sum is the Hecke series of a nonholomorphic Cusp form.Abstract:
The summatory function of $t_j(n^2)$ is estimated, where $H_j(s) = \sum_{n=1}^\infty t_j(n)n^{-s}$ is the Hecke series of a non-holomorphic cusp form. The analogous problem of holomorphic cusp forms is also treated.read more
Citations
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The Riemann zeta function
TL;DR: The Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory, algebra, complex analysis, statistics, as well as in physics.
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Spectral theory of the Riemann zeta-function
TL;DR: In this article, non-Euclidean harmonics and automorphic L-functions have been studied, and an explicit formula has been proposed for harmonics with trace formulas.
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Power sums of Hecke eigenvalues and application
TL;DR: In this article, the authors sharpen some estimates of Rankin on power sums of Hecke eigenvalues, by using Kim & Shahidi's recent results on higher order symmetric powers.
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Quasimodularity and large genus limits of Siegel-Veech constants
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On a variance of Hecke eigenvalues in arithmetic progressions
Yuk-Kam Lau,Lilu Zhao +1 more
TL;DR: In this paper, the authors derived asymptotic formulae for the variance ∑ b = 1 q | ∑ n ≤ X n ≡ b ( mod q ) a ( n ) | 2 when X 1 / 4 + e ≤ q ≤ X 1/2 − e or X 1 1 / 2 − e ≥ 0.
References
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Book
The Riemann Zeta-Function
TL;DR: For Re s = σ > 1, the Riemann zeta function ζ(s) is defined by as discussed by the authors, and it follows from the definition that ζ is an analytic function in the halfplane Re s > 1.
Book
Modular Forms and Functions
TL;DR: In this article, the authors present a general characterization of modular forms, including groups of matrices and bilinear mappings, groups of level 2 and sums of squares, hecke operators and congruence groups.
The Riemann zeta function
TL;DR: The Riemann Zeta function is one of the most studied transcendental functions, having in view its many applications in number theory, algebra, complex analysis, statistics, as well as in physics.
Journal ArticleDOI
Coefficient of Maass forms and the Siegel zero (with an Appendix by Dorian Goldfeld, Jeffrey Hoffstein, and Daniel Lieman)
Jeffrey Hoffstein,Paul Lockhart +1 more
Book
Spectral Theory of the Riemann Zeta-Function
TL;DR: Motohashi as discussed by the authors showed that the Riemann zeta function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta functions.