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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]

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 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 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]

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]

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]

Recently, Mangerel extended the Matomäki–Radziwiłł theorem to a large collection of unbounded multiplicative functions in typical short intervals. In this paper, we combine Mangerel's result with Halász-type result recently established by Granville, Harper and Soundararajan to consider the distribution of a class of multiplicative functions in short intervals. First, we prove cancellation in the sum of the coefficients of the standard L-function of an automorphic irreducible cuspidal representation of GLm\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathrm{GL}_m$$\end{document} over Q\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb{Q}$$\end{document} with unitary central character in typical intervals of length h(logX)c\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$h(\log X)^c$$\end{document} with h=h(X)→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$h = h(X) \rightarrow \infty$$\end{document} and some constant c>0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$c > 0$$\end{document} (under Vinogradov–Korobov zero-free region and GRC). Then we also establish a non-trivial bound for the product of divisor-bounded multiplicative functions with the Liouville function in arithmetic progressions over typical short intervals. [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 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 f be a holomorphic Hecke cusp form with even integral weight k ⩾ 2 for the full modular group, and let χ be a primitive Dirichlet character modulo q. Let Lf (s, χ) be the automorphic L-function attached to f and χ. We study the mean-square estimate of Lf (s, χ) and establish an asymptotic formula. [ABSTRACT FROM AUTHOR]

We extend the axiomatization for detecting and quantifying sign changes of Meher and Murty [24] to sequences of complex numbers. We further generalize this result when the sequence is comprised of the coefficients of an L -function. As immediate applications, we prove that there are sign changes in intervals within sequences of coefficients of GL ( 2 ) holomorphic cusp forms, GL ( 2 ) Maass forms, and GL ( 3 ) Maass forms. Building on [13,14] , we prove that there are sign changes in intervals within sequences of partial sums of coefficients of GL ( 2 ) holomorphic cusp forms and Maass forms. [ABSTRACT FROM AUTHOR]

Abstract Let π and π′ be irreducible cuspidal automorphic representations of GL (픸) and GL (픸), respectively. Denote by a ( n) the nth coefficient of the Dirichlet series of the Rankin-Selberg L-function associated with π and π′. In this paper, we obtain the upper bound for the sum Σ a ( n). [ABSTRACT FROM AUTHOR]

Let f be a Hecke eigencusp form of even integral weight k or Maass cusp form for the full modular group S L 2 ( Z ) . Denote by λ sym m f ( n ) the n th normalized coefficient of the Dirichlet expansion of the m th symmetric power L -function associated to f . In this paper, we establish some bounds for ∑ n ⩽ x λ sym m f ( n ) , ∑ n ⩽ x λ f ( n m ) , which improve the corresponding results of Lau & Lü [8] . [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]

Let { ϕ ( z) : j ≥ 1} be an orthonormal basis of Hecke-Maass cusp forms with Laplace eigenvalue 1/4 + t . For each ϕ ( z), we have the automorphic L-function L( s, sym ϕ ), which is called the symmetric square L-function associated to ϕ . In this paper, we consider the average estimate of L(1/2, sym ϕ ) and prove that, for sufficiently large T, the estimate $$ {\displaystyle {\sum}_{T<{t}_j\le T+M}{\left|L\left(1/2,{\mathrm{sym}}^2{\phi}_j\right)\right|}^2\ll {T}^{1+\varepsilon }M} $$ holds for T ≤ M ≤ T . [ABSTRACT FROM AUTHOR]

In this paper, we define the Euler phi function φ π( n) associated to the automorphic L-function L( s, π), where π is an automorphic irreductible unitary cuspidal representation of GL N(ℚ). Furthermore, we study the asymptotic properties for φ π( n) and prove some results on the summation function ∑ n≤ x φ π( n). [ABSTRACT FROM AUTHOR]

We show that a weak form of the generalized Böcherer conjecture implies multiplicity one for Siegel cusp forms of degree 2. [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]

Let π i , i = 1 , 2 , 3 , be unitary automorphic cuspidal representations of G L 2 (Q A) with Fourier coefficients λ π i (n) . Consider an automorphic representation Π which is equivalent to ∧ 2 (Sym 3 π 1) , π 1 ⊠ π 2 , π 1 ⊠ Sym 2 π 2 , ∧ 2 (π 1 ⊠ π 2) , or π 1 × π 2 × π 3 . Since the Dirichlet series of L (s , Π × Π ~) is known to be complicated, a simpler Dirichlet series ∑ λ (n) n - s is defined and analytically continued in each case, which is closely related to L (s , Π × Π ~) and catches the essence of the underlying functoriality. Asymptotics of ∑ n ≤ x λ (n) are proved. As applications, certain means, variance, and covariances of | λ π i (n) | k for k = 2 , 4 , 6 and | λ π i (n j) | 2 for j = 2 , 3 , 4 are computed. These statistics provide a deep insight of the distribution of the GL(2) Fourier coefficients λ π i (n) . [ABSTRACT FROM AUTHOR]