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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]

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 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]

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 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]

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 extend the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h (log X) c , with h = h (X) → ∞ and where c = c f ≥ 0 is determined by the distribution of { | f (p) | } p in an explicit way. We give three applications. First, we show that the classical Rankin–Selberg-type asymptotic formula for partial sums of | λ f (n) | 2 , where { λ f (n) } n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length h log X , if h = h (X) → ∞ . We also generalize this result to sequences { | λ π (n) | 2 } n , where λ π (n) is the nth coefficient of the standard L-function of an automorphic representation π with unitary central character for G L m , m ≥ 2 , provided π satisfies the generalized Ramanujan conjecture. Second, using recent developments in the theory of automorphic forms we estimate the variance of averages of all positive real moments { | λ f (n) | α } n over intervals of length h (log X) c α , with c α > 0 explicit, for any α > 0 , as h = h (X) → ∞ . Finally, we show that the (non-multiplicative) Hooley Δ -function has average value ≫ log log X in typical short intervals of length (log X) 1 / 2 + η , where η > 0 is fixed. [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]

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 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 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]

We study the values of | L ( 1 , F ) | for Hecke–Maass cusp forms F on S L ( n , Z ) ( n ≥ 3 ) of large Langlands parameters. New unconditional results on the extreme values and conditional results on the size range are derived, which determine precisely the order of magnitude of L ( 1 , F ) . In addition, we enhance the new average estimate toward the Ramanujan Conjecture due to Matz and Templier. An application of the Hecke multiplicativity to the Littlewood–Richardson rule for a product of two Schur polynomials is cultivated. [ABSTRACT FROM AUTHOR]

Let f be a nonzero cusp form of even integral weight k ⩾ 2 on the Hecke congruence subgroup Γ 0 ( N ) with N squarefree. Suppose that the normalized Fourier coefficients λ f ( n ) of f are real. We prove that the first sign change of λ f ( n ) occurs in the range n ≪ ( k N ) 2 + ε . This improves upon the earlier result of Choie and Kohnen [2] . [ABSTRACT FROM AUTHOR]

Assume that α , β are arbitrary arithmetic functions, and that γ is an arithmetic function, which does not correlate with additive characters. By introducing a simple argument, we are able to give a general upper bound for the triple correlation ∑ n ∈ ( X , 2 X ] α ( n ) β ( n − h ) γ ( n + h ) averaging over a short variable h . More precise estimates can be obtained by taking γ to be Fourier coefficients of cusp forms, which improve and streamline the results of Lin and Singh. [ABSTRACT FROM AUTHOR]

Let g be a holomorphic or Maass Hecke newform of level D and nebentypus χD, and let a ( n) be its n-th Fourier coefficient. We consider the sum $${S_1} = \sum {_{X < n \leqslant 2X}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)}$$ and prove that S has an asymptotic formula when β = 1/2 and α is close to $$\pm 2\sqrt {q/D}$$ for positive integer q ≤ X/4 and X sufficiently large. And when 0 < β < 1 and α, β fail to meet the above condition, we obtain upper bounds of S . We also consider the sum $${S_2} = \sum {_{n > 0}{a_g}\left( n \right)e\left( {\alpha {n^\beta }} \right)\phi \left( {n/X} \right)}$$ with ø( x) ∈ C (0,+∞) and prove that S has better upper bounds than S at some special α and β. [ABSTRACT FROM AUTHOR]

Let f and g be weight k holomorphic cusp forms and let S f ( n ) and S g ( n ) denote the sums of their first n Fourier coefficients. Hafner and Ivić [9] proved asymptotics for ∑ n ≤ X | S f ( n ) | 2 and proved that the Classical Conjecture, that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ , holds on average over long intervals. In this paper, we introduce and obtain meromorphic continuations for the Dirichlet series D ( s , S f × S g ) = ∑ S f ( n ) S g ( n ) ‾ × n − ( s + k − 1 ) and D ( s , S f × S g ‾ ) = ∑ n S f ( n ) S g ( n ) n − ( s + k − 1 ) . We then prove asymptotics for the smoothed second moment sums ∑ S f ( n ) S g ( n ) ‾ e − n / X , giving a smoothed generalization of [9] . We also attain asymptotics for analogous sums of normalized Fourier coefficients. Our methodology extends to a wide variety of weights and levels, and comparison with [4] indicates very general cancellation between the Rankin–Selberg L -function L ( s , f × g ) and convolution sums of the coefficients of f and g . In forthcoming works, the authors apply the results of this paper to prove the Classical Conjecture on | S f ( n ) | 2 is true on short intervals, and to prove sign change results on { S f ( n ) } n ∈ N . [ABSTRACT FROM AUTHOR]

Let f be a weight k holomorphic cusp form of level one, and let S f ( n ) denote the sum of the first n Fourier coefficients of f . In analogy with Dirichlet's divisor problem, it is conjectured that S f ( X ) ≪ X k − 1 2 + 1 4 + ϵ . Understanding and bounding S f ( X ) has been a very active area of research. The current best bound for individual S f ( X ) is S f ( X ) ≪ X k − 1 2 + 1 3 ( log ⁡ X ) − 0.1185 from Wu [17] . Chandrasekharan and Narasimhan [2] showed that the Classical Conjecture for S f ( X ) holds on average over intervals of length X . Jutila [10] improved this result to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 3 4 + ϵ . Building on the results and analytic information about ∑ | S f ( n ) | 2 n − ( s + k − 1 ) from our recent work [9] , we further improve these results to show that the Classical Conjecture for S f ( X ) holds on average over short intervals of length X 2 3 ( log ⁡ X ) 1 6 . [ABSTRACT FROM AUTHOR]

Under a slightly better zero-free region of the corresponding modular L -function, we get a very small bound for the size of first sign change of Hecke eigenvalues for the classical modular forms. When it comes to the prime argument, we derive, for almost all primitive forms, a small bound for the first sign change on prime numbers. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study products of Fourier coefficients of an elliptic cusp form, () for a fixed positive integer r, concerning both non-vanishing and non-negativity. [ABSTRACT FROM AUTHOR]

Let f be a holomorphic Hecke eigenform of weight k for the modular group Γ = SL (Z) and let λ ( n) be the n-th normalized Fourier coefficient. In this paper, by a new estimate of the second integral moment of the symmetric square L-function related to f, the estimate is established, which improves the previous result. [ABSTRACT FROM AUTHOR]

In this paper, we study the exponential sums involving Fourier coefficients of cusp forms for SL ( 2 , Z ) over primes. We are able to establish some new results. In addition, similar results have been obtained for the coefficients of symmetric-square lifts. [ABSTRACT FROM AUTHOR]

For the normalized Fourier coefficients of Maass cusp forms λ( n) and the normalized Fourier coefficients of holomorphic cusp forms a( n), we give the bound of $\sum\nolimits_{m_1^2 + m_2^2 + m_3^2 \leqslant x} {\lambda (m_1^2 + m_2^2 + m_3^2 )\Lambda (m_1^2 + m_2^2 + m_3^2 )} $ and $\sum\nolimits_{m_1^2 + m_2^2 + m_3^2 \leqslant x} {a(m_1^2 + m_2^2 + m_3^2 )\Lambda (m_1^2 + m_2^2 + m_3^2 )} $. [ABSTRACT FROM AUTHOR]

We formulate an analogue of the Ichino–Ikeda conjectures for the Whittaker–Fourier coefficients of automorphic forms on quasi-split reductive groups. This sharpens the conjectures of Sakellaridis–Venkatesh in the case at hand. [ABSTRACT FROM AUTHOR]

We characterize all cusp forms among the degree two Siegel modular forms by the growth of their Fourier coefficients. We also give a similar result for Jacobi forms over the group . [ABSTRACT FROM AUTHOR]

Let λf( n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f( z) ∈ Sk(Γ). In this paper, we established nontrivial estimates for $$\sum\limits_{n \leqslant x} {\lambda _f \left( {n^i } \right)\lambda _f \left( {n^j } \right),}$$ where 1 ≤ i < j ≤ 4. [ABSTRACT FROM AUTHOR]