A twist of the Gauss circle problem by holomorphic cusp forms

2021 ◽  
Vol 8 (1) ◽  
Author(s):  
Ratnadeep Acharya
Keyword(s):  
Cusp Forms ◽  
Circle Problem ◽  
Holomorphic Cusp Forms ◽  
Gauss Circle Problem

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 On the Distribution of Periods of Holomorphic Cusp Forms and Zeroes of Period Polynomials

2020 ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Behavior ◽  
Method Of Moments ◽  
Cusp Form ◽  
Joint Distribution ◽  
Random Variables ◽  
Kloosterman Sums ◽  
Limiting Distribution ◽  
Independent Random Variables ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

Abstract In this paper, we determine the limiting distribution of the image of the Eichler–Shimura map or equivalently the limiting joint distribution of the coefficients of the period polynomials associated to a fixed cusp form. The limiting distribution is shown to be the distribution of a certain transformation of two independent random variables both of which are equidistributed on the circle $\mathbb{R}/\mathbb{Z}$, where the transformation is connected to the additive twist of the cuspidal $L$-function. Furthermore, we determine the asymptotic behavior of the zeroes of the period polynomials of a fixed cusp form. We use the method of moments and the main ingredients in the proofs are additive twists of $L$-functions and bounds for both individual and sums of Kloosterman sums.

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.

ON THE DIMENSION OF THE SPACE OF CUSP FORMS OF OCTAHEDRAL TYPE

2010 ◽  
Vol 06 (04) ◽  
pp. 767-783 ◽  
Author(s):  
SATADAL GANGULY
Keyword(s):  
Galois Representations ◽  
Character Formula ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms ◽  
Weight One ◽  
Space Of Cusp Forms

For a prime q ≡ 3 ( mod 4) and the character [Formula: see text], we consider the subspace of the space of holomorphic cusp forms of weight one, level q and character χ that is spanned by forms that correspond to Galois representations of octahedral type. We prove that this subspace has dimension bounded by [Formula: see text] upto multiplication by a constant that depends only on ε.

scholarly journals Central values of additive twists of cuspidal L-functions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Asbjørn Christian Nordentoft
Keyword(s):  
Asymptotic Formula ◽  
Cusp Form ◽  
Dirichlet Character ◽  
Cusp Forms ◽  
Modular Symbols ◽  
Central Values ◽  
Arithmetic Distribution ◽  
Holomorphic Cusp Forms ◽  
Normally Distributed

Abstract Additive twists are important invariants associated to holomorphic cusp forms; they encode the Eichler–Shimura isomorphism and contain information about automorphic L-functions. In this paper we prove that central values of additive twists of the L-function associated to a holomorphic cusp form f of even weight k are asymptotically normally distributed. This generalizes (to k ≥ 4 {k\geq 4} ) a recent breakthrough of Petridis and Risager concerning the arithmetic distribution of modular symbols. Furthermore, we give as an application an asymptotic formula for the averages of certain “wide” families of automorphic L-functions consisting of central values of the form L ⁢ ( f ⊗ χ , 1 / 2 ) {L(f\otimes\chi,1/2)} with χ a Dirichlet character.

Sign in / Sign up

Export Citation Format

Share Document