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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 Hk denote the space of holomorphic Hecke cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Let λf (n), λg(n), and λh(n) be the nth Hecke-normalized Fourier coefficients of three distinct holomorphic Hecke cusp forms f(z), g(z), h(z) ∈ H k i ′ , i ′ = 1 , 2 , 3 , respectively. In this paper, we establish an asymptotic formula for the summatory function (under suitable conditions)Σn≤x λ f 2 (ni) λ g 2 (nj), which generalizes the results of Lü [17]. As an application, we also consider several averages of absolute values of Fourier coefficients of holomorphic Hecke cusp forms of the type Σn≤x |λf (n)λg(ni)| and Σn≤x|λf (n)λg(ni)λh(nj)|, which also generalizes the work of Lü [17]. [ABSTRACT FROM AUTHOR]

We investigate the behaviour of orthogonal non-holomorphic Eisenstein series at their harmonic points by using theta lifts. In the case of singular weight, we show that the orthogonal non-holomorphic Eisenstein series that can be written as a theta lift have a simple pole at s = 1 whose residues yield holomorphic orthogonal modular forms that are Eisenstein series on the boundary. Moreover, we will investigate the image of this construction and give sufficient conditions for the surjectivity. [ABSTRACT FROM AUTHOR]

Suppose ∑ n = 1 ∞ a (n) n - s be a Dirichlet series in the Selberg class of degree d and let E(x) be the arithmetical error term of ∑ n ⩽ x a (n) . By the truncated Tong-type formula of E(x), we can get two kinds of the mean square estimates of E(x) in short intervals of Jutila's type. Using the estimates, we are able to improve some previous results established by Lü and Wang [9]. [ABSTRACT FROM AUTHOR]

We study sign changes and non-vanishing of a certain double sequence of Fourier coefficients of cusp forms of half-integral weight. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. In this paper we compute the Fourier coefficients of the Siegel Eisenstein series of degree 2, level p with the trivial or the quadratic character, associated to a certain cusp. For that we need to define the p-factor of the special type of Siegel series with character. [ABSTRACT FROM AUTHOR]

We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight g 2 and genus g. When g = 4 , the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices. [ABSTRACT FROM AUTHOR]

Duke and Imamoglu showed that the Saito–Kurokawa lift for Siegel modular forms can in an elegant manner be obtained from a converse theorem by Imai using spectral analysis of the hyperbolic Laplacian. In an earlier paper we gave a simplified approach without any appeal to spectral analysis. Here we want to show, that this generalizes to some orthogonal groups of signature (2 , m + 2) with m > 2 . [ABSTRACT FROM AUTHOR]

Let k ≥ 2 and N be positive integers and let χ be a Dirichlet character modulo N. Let f(z) be a modular form in M k (Γ 0 (N) , χ) . Then we have a unique decomposition f (z) = E f (z) + S f (z) , where E f (z) ∈ E k (Γ 0 (N) , χ) and S f (z) ∈ S k (Γ 0 (N) , χ) . In this paper, we give an explicit formula for E f (z) in terms of Eisenstein series whose coefficients are sum of divisors function. Then we apply our result to certain families of eta quotients and to representations of positive integers by 2k–ary positive definite quadratic forms in order to give an alternative version of Siegel's formula for the weighted average number of representations of an integer by quadratic forms in the same genus. Our formula for the latter is in terms of sum of divisors function and does not involve computation of local densities. [ABSTRACT FROM AUTHOR]

In this paper, we consider twisted Dirichlet series attached to quasimodular forms, study their analytic properties, and prove an analogue of Weil's converse theorem for quasimodular forms over congruence subgroups. We also give some applications of our results to a certain q-series and sign changes of the Fourier coefficients of quasimodular forms. [ABSTRACT FROM AUTHOR]

We consider a special class of unipotent periods for automorphic forms on a finite cover of a reductive adelic group G (A K) , which we refer to as Fourier coefficients associated to the data of a 'Whittaker pair'. We describe a quasi-order on Fourier coefficients, and an algorithm that gives an explicit formula for any coefficient in terms of integrals and sums involving higher coefficients. The maximal elements for the quasi-order are 'Levi-distinguished' Fourier coefficients, which correspond to taking the constant term along the unipotent radical of a parabolic subgroup, and then further taking a Fourier coefficient with respect to a K -distinguished nilpotent orbit in the Levi quotient. Thus one can express any Fourier coefficient, including the form itself, in terms of higher Levi-distinguished coefficients. In companion papers we use this result to determine explicit Fourier expansions of minimal and next-to-minimal automorphic forms on split simply-laced reductive groups, and to obtain Euler product decompositions of certain Fourier coefficients. [ABSTRACT FROM AUTHOR]

Suppose that f is a primitive Hecke eigenform or a Mass cusp form for Γ0(N) with normalized eigenvalues λf (n) and let X > 1 be a real number. We consider the sum and show that for every k ⩾ 1 and ε > 0. The same problem was considered for the case N = 1, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for k ⩾ 5. Since the result is valid for arbitrary level, we obtain, as a corollary, estimates on sums of the form , where the sum involves restricted coefficients of some suitable half integral weight modular forms. [ABSTRACT FROM AUTHOR]

Given a complex number a, we prove that there is at most one normalized Hecke eigenform of level one whose third Fourier coefficient is given by a, assuming that the characteristic polynomials of the Hecke operators T 3 are irreducible. This generalizes a result of Vilardi and Xue on the second Fourier coefficients. Under the assumption on irreducibility of characteristic polynomials of T 2 and T 3 , we further show that there is at most one normalized eigenform of level one whose second or third Fourier coefficient is given by a. [ABSTRACT FROM AUTHOR]

We prove power-saving bounds for general Kloosterman sums on Sp (4) associated to all Weyl elements via a stratification argument coupled with p-adic stationary phase methods. We relate these Kloosterman sums to the Fourier coefficients of Sp (4) Poincare series. [ABSTRACT FROM AUTHOR]

In this note, we give a detailed proof of an asymptotic for averages of coefficients of a class of degree three L-functions which can be factorized as a product of a degree one and a degree two L-functions. We emphasize that we can break the 1/2-barrier in the error term, and we get an explicit exponent. Our proof relies on methods from the theory of exponential sums. [ABSTRACT FROM AUTHOR]

We prove a Voronoi–Oppenheim summation formula for divisor functions associated with Gaussian integers. This formula is a direct generalization of Oppenheim's summation formula for classical divisor functions. To prove the formula we construct an Eisenstein series and study its properties. Our method of proof is similar to Beineke and Bump's proof of the classical Oppenheim summation formula. [ABSTRACT FROM AUTHOR]

We derive an explicit formula for the trace of an arbitrary Hecke operator on spaces of twist-minimal holomorphic cusp forms with arbitrary level and character, and weight at least 2. We show that this formula provides an efficient way of computing Fourier coefficients of basis elements for newform or cusp form spaces. This work was motivated by the development of a twist-minimal trace formula in the non-holomorphic case by Booker, Lee and Strömbergsson, as well as the presentation of a fully generalised trace formula for the holomorphic case by Cohen and Strömberg. [ABSTRACT FROM AUTHOR]

In this paper, we solve the Rankin–Selberg problem. That is, we break the well known Rankin–Selberg's bound on the error term of the second moment of Fourier coefficients of a GL (2) cusp form (both holomorphic and Maass), which remains its record since its birth for more than 80 years. We extend our method to deal with averages of coefficients of L-functions which can be factorized as a product of a degree one and a degree three L-functions. [ABSTRACT FROM AUTHOR]

A natural variant of Lehmer's conjecture that the Ramanujan τ -function never vanishes asks whether, for any given integer α , there exist any n ∈ Z + such that τ (n) = α . A series of recent papers excludes many integers as possible values of the τ -function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for τ (n) . We synthesize these results and methods to prove that if 0 < α < 100 and α ∉ T : = { 2 k , - 24 , - 48 , - 70 , - 90 , 92 , - 96 } , then τ (n) ≠ α for all n > 1 . Moreover, if α ∈ T and τ (n) = α , then n is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that τ (n) > 100 for all n > 2 . [ABSTRACT FROM AUTHOR]

We examine an arithmetical function defined by recursion relations on the sequence {f(pk)}k∈ℕ and obtain sufficient condition(s) for the sequence to change sign infinitely often. As an application we give criteria for infinitely many sign changes of Chebyshev polynomials and that of sequence formed by the Fourier coefficients of a cusp form. [ABSTRACT FROM AUTHOR]

In this article, we have considered the problem of first sign change of Fourier coefficients of a primitive cusp form at a sparse sequence, where elements of the sparse sequence can be written as a sum of two squares. Following the methods of Kohnen–Sengupta, we give an effective upper bound on elements of such a sparse sequence to have first sign change. [ABSTRACT FROM AUTHOR]

Let π be a self-dual irreducible cuspidal automorphic representation of GL2( A ℚ ) with trivial central character. Its Hecke eigenvalue ⁁π (n) is a real multiplicative function in n. We show that λπ (n) < 0 for some n < < Q π 2 / 5 , where Qπ denotes (a special value of) the analytic conductor. The value 2 5 is the first explicit exponent for Hecke-Maass newforms. [ABSTRACT FROM AUTHOR]

Let λ π (1 , ... , 1 , n) be the normalized Fourier coefficients of an even Hecke–Maass form π for S L (m , Z) with m ≥ 3 , and r 3 (n) = # { (n 1 , n 2 , n 3) ∈ Z 3 : n = n 1 2 + n 2 2 + n 3 2 } . In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients λ π (1 , ... , 1 , n) and r 3 (n) , which improves previous results (even under the generalized Ramanujan conjecture). [ABSTRACT FROM AUTHOR]

The Bourgain-Sarnak-Ziegler (BSZ) criterion was initially established for an arbitrary multiplicative function a(n) satisfying |a(n)| ⩽ 1. Recently, Cafferata et al. (2020) showed that the condition |a(n)| ⩽ 1 can be replaced by a(p) 1 to some extent. In this paper, we formulate and prove a further extension of the BSZ criterion, in which the restriction a(p) ⩽ 1 is removed. As applications, we use it together with the analytic theory of automorphic L-functions to prove that there exist some cancellations in the sequence {λπ(n)e(nkα)}n⩽1 on GLm and the Möbius function is disjoint from this sequence. [ABSTRACT FROM AUTHOR]

In this survey article, we explain how false theta functions can be embedded into a modular framework and show some of the applications of this modularity. [ABSTRACT FROM AUTHOR]

In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and r s , k (n) , the number of representations of a positive integer n as a sum of s k -th positive integral powers, based on the recently proved Main Conjecture in Vinogradov's Mean Value Method. [ABSTRACT FROM AUTHOR]

Let ϕ (z) be a primitive Hecke–Maass cusp forms with Laplace eigenvalue 1 4 + t 2 . Denote by L (s , sym m ϕ) the m-th symmetric power L-function associated to ϕ and by λ sym m ϕ (n) the n-th coefficient of the Dirichlet expansion of L (s , sym m ϕ) . For any nonzero integer ℓ we prove ∑ n ⩽ x λ ϕ (n) λ ϕ (n + ℓ) ≪ ϕ , ℓ x (log x) 0.187 (x ⩾ 3). This improves Holowinsky's corresponding result, which requires 1 6 in place of 0.187. for all x ⩾ 3 . Further assuming that L (s , sym 10 ϕ) and L (s , sym 12 ϕ) are automorphic cuspidal, we obtain a conditional generalization to the symmetric square case: ∑ n ⩽ x λ sym 2 ϕ (n) λ sym 2 ϕ (n + ℓ) ≪ ϕ , ℓ x (log x) 0.196 (x ⩾ 3). [ABSTRACT FROM AUTHOR]

Let f, g be distinct normalized Hecke eigenforms of weights k 1 , k 2 lying in the subspace of newforms with Fourier coefficients { n (k 1 - 1) / 2 λ f (n) } n ∈ N and { n (k 2 - 1) / 2 λ g (n) } n ∈ N respectively. For such newforms f, g of CM type and primes p, we study the natural density of the set S = { p | λ f (p) = λ g (p) }. We show that the upper natural density of S is ≤ 3 / 4 if f ≠ g and it is equal to 1/2 when f and g have different weights and have the same associated CM (quadratic) field. Further, f and g have different associated CM (quadratic) fields if and only if the natural density of S is 1/4. When at least one of f, g is a non-CM form, we study the natural density of the sets S + (x , α) = { p ≤ x | θ f (p) + θ g (p) = α } and S - (x , β) = { p ≤ x | θ f (p) - θ g (p) = β } where θ f (p) , θ g (p) ∈ [ 0 , π ] are the angles associated to the p-th Hecke eigen values of f, g respectively and α ∈ [ 0 , 2 π ] , β ∈ [ - π , π ] . In this case, we show that S + (x , α) and S - (x , β) have natural density zero when f and g are distinct and not twists of each other. Finally, we establish an explicit link between the elements of these sets and sign changes of Fourier coefficients at prime powers which allows us to improve a number of existing results in this set-up. [ABSTRACT FROM AUTHOR]

Let R 1 (n) , R 2 (n) denote the numbers of representations of a positive integer n by the quaternary quadratic forms g 1 (x 1 , x 2 , x 3 , x 4) = 2 (x 1 2 + x 1 x 2 + x 2 2) + 2 x 1 x 3 + x 1 x 4 + x 2 x 3 + 2 x 2 x 4 + 2 (x 3 2 + x 3 x 4 + x 4 2) , g 2 (x 1 , x 2 , x 3 , x 4) = 8 (x 1 2 + x 2 2) + x 3 2 + x 4 2 , respectively, where x 1 , x 2 , x 3 and x 4 are integers. In this paper, we establish the asymptotic formulae for the sums ∑ n ≤ x R i (n) and ∑ n ≤ x R i 2 (n) for i = 1 , 2 . [ABSTRACT FROM AUTHOR]

We construct a family of modular forms from harmonic Maass Jacobi forms by considering their Taylor expansion and using the method of holomorphic projection. As an application we present a certain type Hurwitz class relations which can be viewed as a generalization of Mertens' result in M. H. Mertens (2016). [ABSTRACT FROM AUTHOR]