scholarly journals An explicit Hecke's bound and exceptions of even unimodular quadratic forms

2002 ◽  
Vol 65 (2) ◽  
pp. 231-238
Author(s):  
Kok Seng Chua
Keyword(s):  
Quadratic Forms ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Holomorphic Cusp Forms

We prove an explicit Hecke's bound for the Fourier coefficients of holomorphic cusp forms for SL2(Z) and apply it to derive effectively computable constants c (m) for each dimension m, divisible by 8, for which every even integer is always represented by every even unimodular form of m variables.

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.

On the typical size and cancellations among the coefficients of some modular forms

2018 ◽  
Vol 166 (1) ◽  
pp. 173-189
Author(s):  
FLORIAN LUCA ◽  
MAKSYM RADZIWIŁŁ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Binary Quadratic Form ◽  
Cusp Forms ◽  
Real Numbers ◽  
Standard Argument ◽  
Typical Size ◽  
Holomorphic Cusp Forms ◽  
Weak Hypothesis ◽  
Almost All

AbstractWe obtain a nontrivial upper bound for almost all elements of the sequences of real numbers which are multiplicative and at the prime indices are distributed according to the Sato–Tate density. Examples of such sequences come from coefficients of severalL-functions of elliptic curves and modular forms. In particular, we show that |τ(n)| ⩽n11/2(logn)−1/2+o(1)for a set ofnof asymptotic density 1, where τ(n) is the Ramanujan τ function while the standard argument yields log 2 instead of −1/2 in the power of the logarithm. Another consequence of our result is that in the number of representations ofnby a binary quadratic form one has slightly more than square-root cancellations for almost all integersn.In addition, we obtain a central limit theorem for such sequences, assuming a weak hypothesis on the rate of convergence to the Sato–Tate law. For Fourier coefficients of primitive holomorphic cusp forms such a hypothesis is known conditionally and might be within reach unconditionally using the currently established potential automorphy.

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.

scholarly journals A relation between Fourier coefficients of holomorphic cusp forms and exponential sums

2009 ◽  
Vol 86 (100) ◽  
pp. 97-105 ◽  
Author(s):  
Anne-Maria Ernvall-Hytönen
Keyword(s):  
Fourier Coefficients ◽  
Exponential Sums ◽  
Cusp Forms ◽  
Lehmer Conjecture ◽  
Holomorphic Cusp Forms

We consider certain specific exponential sums related to holomorphic cusp forms, give a reformulation for the Lehmer conjecture and prove that certain exponential sums of Fourier coefficients of holomorphic cusp forms contain information on other similar non-overlapping exponential sums. Also, we prove an Omega result for short sums of Fourier coefficients.

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