Search

Fix an integer κ ≥ 2 . Let P ≥ 2 be a prime, and F be the symmetric-square lift of a Hecke newform f ∈ S κ * (P) . We study the exponential sum L F (α) = ∑ n ∼ N A F (n , 1) e (n α) by implementing an average over a family in such a way to investigate the best possible magnitude of the level aspect bound for L F (α) . We prove a uniform bound with respect to any α ∈ R and the level parameter P , and present that there exist certain forms with fairly strong oscillations in L F (α) , if the associated level of f is allowed to vary. As applications, we consider the shifted convolution sums for GL (3) × GL (d) , for any d ≥ 2 , in a family as well as theWaring-Goldbach problem associated to Fourier coefficients of SL (3 , Z) -Maa β forms. [ABSTRACT FROM AUTHOR]

Let f and g be two different holomorphic cusp froms or Maass cusp forms for the full modular group S L (2 , Z) . We are interested in coefficients of Rankin–Selberg L-functions, and establish some bounds for ∑ n ≤ x λ sym i f × sym j g (n) , ∑ n ≤ x λ f (n i) λ g (n j) , ∑ n ≤ x | λ sym i f × sym j g (n) | , ∑ n ≤ x | λ f (n i) λ g (n j) | , and ∑ n ≤ x max { | λ sym i f × sym j g (n) | 2 φ , | λ sym i f × sym j g (n + h) | 2 φ } , where φ > 0 and h is a fixed positive integer. [ABSTRACT FROM AUTHOR]

We consider the k-th Riesz mean for the coefficients of the Rankin–Selberg L-function related to the Godement–Jacquet L-function with respect to S L (n , Z) and establish an asymptotic formula with a good error term for k ≥ k 0 (n) where k 0 (n) is a positive integer depending only on n. [ABSTRACT FROM AUTHOR]

For fixed positive integers k and N, there are only finitely many holomorphic eta quotients of weight k for the congruence subgroup \(\Gamma_{0}(N)\). In this article, we obtain a similar finiteness result for holomorphic generalized eta quotients of weight k on \(\Gamma_1(p)\) where k is a positive integer and p is a prime number. We also obtain a criterion for two holomorphic generalised eta quotients which represent the same one. [ABSTRACT FROM AUTHOR]

We consider the simultaneous sign changes of normalized Fourier coefficients associated to two distinct Hecke eigenforms supported at positive integers represented by a primitive integral binary quadratic form with negative discriminant whose class number is 1. We provide a quantitative result for the number of sign changes of such sequence in the interval (x , 2 x ] for sufficiently large x . [ABSTRACT FROM AUTHOR]

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by λ f (n) . Let g(z) be another distinct primitive holomorphic cusp form of even integral weight ℓ for the full modular group. In this paper, we establish that the set { p | λ f (p j) < λ g (p j) } with 1 ≤ j ≤ 8 has analytic density at least 1 16 [ j + 1 2 ] 2 , and the set { p | λ f (p j) 2 < λ g (p j) 2 } with 1 ≤ j ≤ 4 has analytic density at least 1 4 j (j + 1) 2 . [ABSTRACT FROM AUTHOR]

We study a square mean value problem and two general divisor problems related to Hecke eigenvalues of classical holomorphic cusp forms and classical Maass cusp forms, respectively. We improve previous results. [ABSTRACT FROM AUTHOR]

We study the mean value estimate of coefficients of product L-function related to a primitive holomorphic cusp form f(z) of weight k for the full modular group SL2(Z). Upper bounds and asymptotic formulas are established. [ABSTRACT FROM AUTHOR]

Let f( z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke eigenvalue) by λ( n). Let d( n) be the Dirichlet divisor function. In this paper, we establish that where $${0 < \delta_{j} < 1}$$ , $${0 < \delta_{i,j}\leqq 1}$$ ( $${\delta_{i,j} = 1}$$ if and only if i = j), and . [ABSTRACT FROM AUTHOR]