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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]

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 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]

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]

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 { 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]