On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms

2011 ◽  
Vol 63 (3) ◽  
pp. 634-647 ◽  
Author(s):  
Guangshi Lü
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

Abstract Let be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let and be the n-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms , respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.(i)For any , we have(ii)If , then for any , we haveIf , then for any , we haveIf and , then for any , we havewhere P(x) is a polynomial of degree 3.

Higher Moments of Fourier Coefficients of Cusp Forms

2015 ◽  
Vol 58 (3) ◽  
pp. 548-560
Author(s):  
Guangshi Lü ◽  
Ayyadurai Sankaranarayanan
Keyword(s):  
Modular Group ◽  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Cusp Forms ◽  
Higher Moments ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

AbstractLet Sk(Γ) be the space of holomorphic cusp forms of even integral weight k for the full modular group SL(z, ℤ). Let be the n-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms , and h(z) ∊ Sk3 (Γ), respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as twisted by the arithmetic function λf(n).

scholarly journals Large sums of Hecke eigenvalues of holomorphic cusp forms

2019 ◽  
Vol 31 (2) ◽  
pp. 403-417
Author(s):  
Youness Lamzouri
Keyword(s):  
Cusp Form ◽  
Riemann Hypothesis ◽  
Modular Group ◽  
Fourier Coefficients ◽  
Automorphic Forms ◽  
Character Sums ◽  
Cusp Forms ◽  
Hecke Eigenvalues ◽  
Sign Change ◽  
Holomorphic Cusp Forms

AbstractLet f be a Hecke cusp form of weight k for the full modular group, and let {\{\lambda_{f}(n)\}_{n\geq 1}} be the sequence of its normalized Fourier coefficients. Motivated by the problem of the first sign change of {\lambda_{f}(n)}, we investigate the range of x (in terms of k) for which there are cancellations in the sum {S_{f}(x)=\sum_{n\leq x}\lambda_{f}(n)}. We first show that {S_{f}(x)=o(x\log x)} implies that {\lambda_{f}(n)<0} for some {n\leq x}. We also prove that {S_{f}(x)=o(x\log x)} in the range {\log x/\log\log k\to\infty} assuming the Riemann hypothesis for {L(s,f)}, and furthermore that this range is best possible unconditionally. More precisely, we establish the existence of many Hecke cusp forms f of large weight k, for which {S_{f}(x)\gg_{A}x\log x}, when {x=(\log k)^{A}}. Our results are {\mathrm{GL}_{2}} analogues of work of Granville and Soundararajan for character sums, and could also be generalized to other families of automorphic forms.

Fourier Coefficients of Modular Forms of Half-Integral Weight in Arithmetic Progressions

2020 ◽  
Author(s):  
Corentin Darreye
Keyword(s):  
Gaussian Distribution ◽  
Modular Forms ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Arithmetic Progressions ◽  
Cusp Forms ◽  
Half Integral Weight ◽  
Mixed Distribution ◽  
Integral Weight ◽  
Holomorphic Cusp Forms

Abstract We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry–Ganguly–Kowalski–Michel and Kowalski–Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution that comes from the asymptotic mixed distribution of Salié sums.

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

scholarly journals Petersson products of bases of spaces of cusp forms and estimates for Fourier coefficients

2018 ◽  
Vol 14 (08) ◽  
pp. 2277-2290 ◽  
Author(s):  
Rainer Schulze-Pillot ◽  
Abdullah Yenirce
Keyword(s):  
Cusp Form ◽  
Fourier Coefficients ◽  
Orthogonal Basis ◽  
Cusp Forms ◽  
Integral Weight ◽  
Space Of Cusp Forms

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

scholarly journals A short note on sign changes and non-vanishing of Fourier coefficients of half-integral weight cusp forms

2021 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Fourier Coefficients ◽  
Short Note ◽  
Double Sequence ◽  
Cusp Forms ◽  
Sign Changes ◽  
Half Integral Weight ◽  
Integral Weight

AbstractWe study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight.

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