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Let m ≥ 2 be an integer. Let f be a holomorphic Hecke eigenform of even weight k for the full modular group SL(2, ℤ). Denote by λSymmf (n) the nth normalized Dirichlet coefficient of the corresponding symmetric power L-function L(s, Symm f) related to f. In this paper, we study the average behavior of the second moment of the Dirichlet coefficients λSymmf (n) and establish its asymptotic formula. [ABSTRACT FROM AUTHOR]

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]

Howe and Tan (Bull Am Math Soc 28:1–74, 1993) investigated a degenerate principal series representation of indefinite orthogonal groups O (b + , b -) and explicitly described its composition series. In particular it contains a unique unitarizable irreducible submodule Π , which is isomorphic to a cohomological representation. In this paper we construct orthogonal automorphic forms locally corresponding to Π as theta liftings of holomorphic Mp 2 (R) cusp forms by using the Borcherds' method (Invent Math 132:491–562, 1998). We propose a special choice of Schwartz functions to define the liftings, which yields precise descriptions of their Fourier expansions. [ABSTRACT FROM AUTHOR]

Let Γ be a subgroup of finite index of SL 2 ⁡ (퐙) . We give an analytic criterion for a cuspidal divisor to be torsion in the Jacobian J Γ of the corresponding modular curve X Γ . Our main tool is the explicit description, in terms of modular symbols, of what we call Eisenstein cycles. The latter are representations of relative homology classes over which integration of any holomorphic differential forms vanishes. Our approach relies in an essential way on the specific case Γ ⊂ Γ ⁢ (2) , where we can consider convenient generalized Jacobians instead of J Γ . We relate the Eisenstein classes to the scattering constants attached to Eisenstein series. Finally, we illustrate our approach by considering Fermat curves. [ABSTRACT FROM AUTHOR]

Suppose that x is a sufficiently large number and j ≥ 2 is any integer. Let L (s , sym j f) be the j-th symmetric power L-function associated with the primitive holomorphic cusp form f of weight k for the full modular group SL 2 (Z) . Also, let λ sym j f (n) be the n-th normalized Dirichlet coefficient of L (s , sym j f) . In this paper, we establish asymptotic formulas for sums of Dirichlet coefficients λ sym j f (n) over two sparse sequences of positive integers, which improves previous results. [ABSTRACT FROM AUTHOR]

We define L-series of weakly holomorphic quasimodular forms and we derive functional equations of those L-series. We also prove a converse theorem for weakly holomorphic quasimodular forms. [ABSTRACT FROM AUTHOR]

Let τ denote the Ramanujan tau function. One is interested in possible prime values of τ function. Since τ is multiplicative and τ ⁢ (n) is odd if and only if n is an odd square, we only need to consider τ ⁢ (p 2 ⁢ n) for primes p and natural numbers n ≥ 1 . This is a rather delicate question. In this direction, we show that for any ϵ > 0 and integer n ≥ 1 , the largest prime factor of τ ⁢ (p 2 ⁢ n) , denoted by P ⁢ (τ ⁢ (p 2 ⁢ n)) , satisfies P ⁢ (τ ⁢ (p 2 ⁢ n)) > (log ⁡ p) 1 8 ⁢ (log ⁡ log ⁡ p) 3 8 - ϵ for almost all primes p. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients. [ABSTRACT FROM AUTHOR]

Let f, g be two distinct primitive holomorphic cusp forms of even integral weights k 1 , k 2 for the full modular group Γ = S L (2 , Z) , respectively. In this paper, we are interested in the upper bounds for the integral mean square of error terms involving λ f 2 i 1 (n) , λ f 2 (n j 1) and λ f 2 (n i 2) λ g 2 (n j 2) on average in terms of f, g respectively. Here i 1 ≥ 2 , j 1 ≥ 3 and i 2 , j 2 ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

In this paper, we prove the conjectured order lower bound for the k-th moment of central values of quadratic twisted self-dual GL(3) L-functions for all k ≥ 1, based on our recent work on the twisted first moment of central values in this family of L-functions. [ABSTRACT FROM AUTHOR]

Let H k ∗ be the set of all normalized primitive holomorphic cusp forms of even weight k ≥ 2 for S L 2 (Z) . For f ∈ H k ∗ , let λ sym j f (n) be the nth normalized Fourier coefficient of the jth symmetric power L-function attached to f. In this paper, we establish asymptotic formulas for the power sums of λ sym j f (n) and improve previous results. [ABSTRACT FROM AUTHOR]

In this paper, we study shells of the D 4 lattice with a slight generalization of spherical t-designs due to Delsarte–Goethals–Seidel, namely, the spherical design of harmonic index T (spherical T-design for short) introduced by Delsarte-Seidel. We first observe that, for any positive integer m, the 2m-shell of D 4 is an antipodal spherical { 10 , 4 , 2 } -design on the three dimensional sphere. We then prove that the 2-shell, which is the D 4 root system, is a tight { 10 , 4 , 2 } -design, using the linear programming method. The uniqueness of the D 4 root system as an antipodal spherical { 10 , 4 , 2 } -design with 24 points is shown. We give two applications of the uniqueness: a decomposition of the shells of the D 4 lattice in terms of orthogonal transformations of the D 4 root system, and the uniqueness of the D 4 lattice as an even integral lattice of level 2 in the four dimensional Euclidean space. We also reveal a connection between the harmonic strength of the shells of the D 4 lattice and non-vanishing of the Fourier coefficients of a certain newform of level 2. Motivated by this, congruence relations for the Fourier coefficients are discussed. [ABSTRACT FROM AUTHOR]

Let f be a fixed Maass form for SL3 (ℤ) with Fourier coefficients Af(m, n). Let g be a Maass cusp form for SL2 (ℤ) with Laplace eigenvalue 1 4 + k 2 and Fourier coefficient λg(n), or a holomorphic cusp form of even weight k. Denote by SX(f × g, α, β) a smoothly weighted sum of Af(1, n)λg(n)e(αnβ) for X < n < 2X, where α ≠ 0 and β > 0 are fixed real numbers. The subject matter of the present paper is to prove non-trivial bounds for a sum of SX(f × g, α, β) over g as k tends to ∞ with X. These bounds for average provide insight for the corresponding resonance barriers toward the Hypothesis S as proposed by Iwaniec, Luo, and Sarnak. [ABSTRACT FROM AUTHOR]

We consider the Kudla–Millson theta series associated to a quadratic space of signature (N , N) . By combining a "see-saw" argument with the Siegel–Weil formula, we show that its (regularized) integral along a torus attached to a totally real field of degree N is the diagonal restriction of an Eisenstein series. It allows us to express the Fourier coefficients of the diagonal restriction as intersection numbers, which generalizes a result of Darmon, Pozzi and Vonk to totally real fields. [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]

Let k and n be positive even integers. For a Hecke eigenform h in the Kohnen plus subspace of weight k - n 2 + 1 2 for Γ 0 ⁢ (4) , let I n ⁢ (h) be the Duke–Imamoglu–Ikeda lift of h to the space of cusp forms of weight k for Sp n ⁡ (ℤ) . We then give an estimate of the Fourier coefficients of I n ⁢ (h) . It is better than the usual Hecke bound for the Fourier coefficients of a Siegel cusp form. [ABSTRACT FROM AUTHOR]

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that lim x → ∞ # { 1 ≤ n ≤ x ∣ τ (n) ≠ 0 } x > 1 - 1.15 × 10 - 12 , where τ (n) is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that τ (n) ≠ 0 . [ABSTRACT FROM AUTHOR]

We give a limit theorem with respect to the matrices related to non-backtracking paths of a regular graph. The limit obtained closely resembles the kth moments of the arcsine law. Furthermore, we obtain the asymptotics of the averages of the p m th Fourier coefficients of the cusp forms related to the Ramanujan graphs defined by A. Lubotzky, R. Phillips and P. Sarnak. [ABSTRACT FROM AUTHOR]

Suppose that H ∗ is the set of all primitive cusp forms f of even integral weight k ≥ 2 for the full modular group S L 2 (Z) . In this paper, we establish asymptotic formulas for the second moment of Fourier coefficients of the triple product L-function L (s , f ⊗ f ⊗ f) and the related L-function L (s , sym 2 f ⊗ f) attached to f on average, which improves previous results. [ABSTRACT FROM AUTHOR]

Let f, g be two distinct normalized primitive holomorphic cusp forms of weights k 1 , k 2 and levels N 1 , N 2 , respectively. Denote by λ f (n) and λ g (n) the n-th normalized Fourier coefficients of f and g, respectively. In this paper, we investigate the first change of sequences { λ f (n i) } n ∈ N and { λ f (n j) λ g (n l) } n ∈ N with positive integers i ≥ 2 and j , l ≥ 1 . And we also derive the lower bounds for the sequences { λ f (n i) } n ∈ N , { λ f (n j) λ f (n l) } n ∈ N of the same signs in the interval [1, x], which remains the best possible results in terms of order of magnitude, where i ≥ 2 , j , l ≥ 1 are positive integers. [ABSTRACT FROM AUTHOR]

Let λ f (n) be the n-th normalized Fourier coefficient of primitive holomorphic cusp form of even integral weight k ≥ 2 for the full modular group S L (2 , Z) . In this paper, we first refine related previous results on the l-th power sums of the Fourier coefficient λ f (n) . Then we establish asymptotic formulas of the variance of λ f l (n) . [ABSTRACT FROM AUTHOR]

We link Ramanujan-type congruences, which emerge abundantly in combinatorics, to the Galois- and geometric theory of modular forms. Specifically, we show that Ramanujan-type congruences are preserved by the action of the shallow Hecke algebra, and discover a dichotomy between congruences originating in Hecke eigenvalues and congruences on arithmetic progressions with cube-free periods. The latter provide congruences among algebraic parts of twisted central L -values. We specialize our results to integer partitions, for which we investigate the landmark proofs of partition congruences by Atkin and by Ono. Based on a modulo ℓ analogue of the Maeda conjecture for certain partition generating functions, we conclude that their approach by Hecke operators acting diagonally modulo ℓ on modular forms is indeed close to optimal. This work is enabled by several structure results for Ramanujan-type congruences that we establish. In an extended example, we showcase how to employ them to also benefit experimental work. [ABSTRACT FROM AUTHOR]

We investigate the linear dependence among odd periods of modular forms. We show that if the dimension of the space of cuspforms is greater than or equal to three, then any three odd periods are linearly independent. We also prove an asymptotic result for an arbitrary number of odd periods. The results are achieved by studying cuspidal projections of products of two Eisenstein series. [ABSTRACT FROM AUTHOR]

Let a E (n) be the normalized Frobenius traces. For any multiplicative function f satisfying some mild conditions, we investigate the sum ∑ n = 1 N f (n) a E (n) and obtain a nontrivial bound. We also prove cancelation in the sum of Frobenius traces twisted by the coefficients of automorphic L-function on GLm. [ABSTRACT FROM AUTHOR]

Let Hk denote the space of primitive holomorphic cusp forms of even integral weight k for the full modular group Γ = SL(2, ℤ). Denote by λ sym m f (n) the nth normalized coefficient of the Dirichlet expansion of the mth symmetric power L-function associated to f. In this paper, we establish the asymptotic formulas of sums of pairwise maxima concerning the normalized coefficients of symmetric power L-functions. We also establish similar results for the normalized coefficients of Rankin-Selberg L-functions L(symif × symjf, s) and L(symif × symjg, s) attached to f and g, respecitvely. As applications, we also consider the proportion of the sign changes among the difference of normalized coefficients associated with symmetric power L-functions and Rankin-Selberg L-functions attached to f and g, respectively. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain asymptotics of shifted sums of Hecke eigenvalue squares on average. We show that for X 2 3 + ϵ < H < X 1 - ϵ there are constants B h such that ∑ X ≤ n ≤ 2 ⁢ X λ f ⁢ (n) 2 ⁢ λ f ⁢ (n + h) 2 - B h ⁢ X = O f , A , ϵ ⁢ (X ⁢ (log ⁡ X) - A ) for all but O f , A , ϵ ⁢ (H ⁢ (log ⁡ X) - 3 ⁢ A ) integers h ∈ [ 1 , H ] where { λ f ⁢ (n) } n ≥ 1 are normalized Hecke eigenvalues of a fixed holomorphic cusp form f. Our method is based on the Hardy–Littlewood circle method. We divide the minor arcs into two parts m 1 and m 2 . In order to treat m 2 , we use the Hecke relations, a bound of Miller to apply some arguments from a paper of Matomäki, Radziwiłł and Tao. In order to treat m 1 , we apply Parseval's identity and Gallagher's lemma. [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]

This article deals with various kinds of quantitative results about the comparison between the normalized Hecke eigenvalues of two distinct Siegel cuspidal Hecke eigenforms for the full symplectic group of degree 2 which are not Saito–Kurokawa lifts. We also prove some simultaneous sign change results for their eigenvalues. [ABSTRACT FROM AUTHOR]

In this article, we derive lower bounds for the number of distinct prime divisors of families of non-zero Fourier coefficients of non-CM primitive cusp forms and more generally of non-CM primitive Hilbert cusp forms. In particular, for the Ramanujan Δ-function, we show that, for any ϵ > 0 , there exist infinitely many natural numbers 푛 such that τ ⁢ (p n) has at least 2 (1 - ϵ) ⁢ log ⁡ n log ⁡ log ⁡ n distinct prime factors for almost all primes 푝. This improves and refines the existing bounds. We also study lower bounds for absolute norms of radicals of non-zero Fourier coefficients of modular forms alluded to above. [ABSTRACT FROM AUTHOR]

Let j ≥ 2 be a given integer. Let f be a normalized primitive holomorphic cusp form of even integral weight for the full modular group Γ = S L (2 , Z) . Denote by λ sym j f (n) the nth normalized coefficient of the Dirichlet expansion of the jth symmetric power L-function L (sym j f , s) attached to f. In this paper, we are interested in the average behaviour of the following sum ∑ a 2 + b 2 + c 2 + d 2 ≤ x (a , b , c , d) ∈ Z 4 λ sym j f 2 (a 2 + b 2 + c 2 + d 2) , where x is sufficiently large, which improves and generalizes the recent work of Sharma and Sankaranarayanan [52]. By analogy, we also consider the analogous results for higher moments of normalized Fourier coefficients and the second moment of normalized coefficients of two symmetric power L-functions attached to two distinct cusp forms of the same sequence. [ABSTRACT FROM AUTHOR]

Let f(z) be a holomorphic Hecke eigenform of even weight k for SL(2, ℤ), and denote by L(s, symmf) the corresponding symmetric power L-function associated with f. Denote by ⋋symmf (n) the nth normalized coefficient of L(s, symmf). In this paper, we investigate the sum Σn≤x⋋symmf (n) for m ≥ 2 and get the uniform upper bound extending the previous results. [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 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]

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 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 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]

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]

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]

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]

In this note, we will apply the results of Gross–Zagier, Gross–Kohnen–Zagier and their generalizations to give a short proof that the differences of singular moduli are not units. As a consequence, we obtain a result on isogenies between reductions of CM elliptic curves. [ABSTRACT FROM AUTHOR]

In this paper, we investigate sign changes of Fourier coefficients of half-integral weight cusp forms. In a fixed square class t Z 2 , we investigate the sign changes in the t p 2 -th coefficient as p runs through the split or inert primes over the ring of integers in a quadratic extension of the rationals. We show that infinitely many sign changes occur in both sets of primes when there exists a prime dividing the discriminant of the field which does not divide the level of the cusp form and find an explicit condition that determines whether sign changes occur when every prime dividing the discriminant also divides the level. [ABSTRACT FROM AUTHOR]

Let $\unicode[STIX]{x1D707}(m,n)$ (respectively, $\unicode[STIX]{x1D702}(m,n)$) denote the number of odd-balanced unimodal sequences of size $2n$ and rank $m$ with even parts congruent to $2\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ (respectively, $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$) and odd parts at most half the peak. We prove that two-variable generating functions for $\unicode[STIX]{x1D707}(m,n)$ and $\unicode[STIX]{x1D702}(m,n)$ are simultaneously quantum Jacobi forms and mock Jacobi forms. These odd-balanced unimodal rank generating functions are also duals to partial theta functions originally studied by Ramanujan. Our results also show that there is a single $C^{\infty }$ function in $\mathbb{R}\times \mathbb{R}$ to which the errors to modularity of these two different functions extend. We also exploit the quantum Jacobi properties of these generating functions to show, when viewed as functions of the two variables $w$ and $q$ , how they can be expressed as the same simple Laurent polynomial when evaluated at pairs of roots of unity. Finally, we make a conjecture which fully characterizes the parity of the number of odd-balanced unimodal sequences of size $2n$ with even parts congruent to $0\!\!\hspace{0.6em}{\rm mod}\hspace{0.2em}4$ and odd parts at most half the peak. [ABSTRACT FROM AUTHOR]

We prove a Green–Tao type theorem for multiplicative functions. [ABSTRACT FROM AUTHOR]

We discuss the generalizations of the concept of Chebyshev's bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev's bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev's bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set {x ≥ 2: ∑p≤xλƒ(p) ≥ 0} for coefficients (λƒ(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known. [ABSTRACT FROM AUTHOR]

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$ th Hecke eigenvalue of  $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author ['Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms', Bull. Aust. Math. Soc.92 (2015), 195–204]. [ABSTRACT FROM AUTHOR]

By elaborating a two-dimensional Selberg sieve with asymptotics and equidistributions of Kloosterman sums from ℓ -adic cohomology, as well as a Bombieri–Vinogradov type mean value theorem for Kloosterman sums in arithmetic progressions, it is proved that for any given primitive Hecke–Maass cusp form of trivial nebentypus, the eigenvalue of the n-th Hecke operator does not coincide with the Kloosterman sum Kl (1 , n) for infinitely many squarefree n with at most 100 prime factors. This provides a partial negative answer to a problem of Katz on modular structures of Kloosterman sums. [ABSTRACT FROM AUTHOR]

Let g, h be two arithmetic functions. In this paper, we study zero-free intervals of polynomials of the type P n g , h (x) := x h (n) ∑ k = 1 n g (k) P n − k g , h (x) , P 0 g , h (x) = 1. We use a difference equation approach. Let g (n) = σ (n) be the divisor sum function and h (n) the identity function, then the zeros of the polynomials dictate the vanishing properties of the coefficients of all powers of the Dedekind eta function. This is directly related to the Lehmer conjecture on the non-vanishing of the Ramanujan τ-function. Let g (n) be the identity function. Then, the polynomials are related to generalized Laguerre polynomials for h (n) = n and to Chebyshev polynomials of the second kind for h (n) = 1. The paper is furnished with several interesting examples. We found polynomials with decreasing minimal real zeros. We provide a simple arithmetic function g to approximate the minimal real zeros attached to σ. [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]

In this article, we investigate the sign changes of the sequence of coefficients at sum of two squares where the coefficients are the Fourier coefficients of the normalized Hecke eigen cusp form for the full modular group. We provide the quantitative result for the number of sign changes of the sequence in a small interval. [ABSTRACT FROM AUTHOR]

We prove nontrivial estimates for linear sums of squares of Fourier coefficients of holomorphic and Maass cusp forms twisted by additive characters. For holomorphic forms f, we show that if | α - a / q | ≤ 1 / q 2 with (a , q) = 1 , then for any ε > 0 , ∑ n ⩽ X λ f (n) 2 e (n α) ≪ f , ε X 4 5 + ε for X 1 5 ≪ q ≪ X 4 5. Moreover, for any ε > 0 , there exists a set S ⊂ (0 , 1) with μ (S) = 1 such that for every α ∈ S , there exists X 0 = X 0 (α) such that the above inequality holds true for any α ∈ S and X ⩾ X 0 (α). A weaker bound for Maass cusp forms is also established. [ABSTRACT FROM AUTHOR]