The Average Estimates For Coefficients Of Rankin–Selberg L-Functions

The automorphic L-function is one of the most important subjects in analytic number theory.In this paper,we investigate the asymptotic behavior of the coefficients of RankinSelberg L-functions,and generalize the previous results.Let Hk*be the set of all normalized Hecke primitive holomorphic cusp forms of even integral weight k for the full modular group.Denote by λf(n)the n-th Fourier coefficients of f∈Hk*.For f∈Hk*,g ∈ Hk’*,in Chapter 3,we apply the pair-Sato-Tate conjecture to establish some results for sums of absolute value of coefficients of L(symi f × symjg,s).Next,we consider the above sums in short intervals.We shall follow results on short sums of multiplicative functions to obtain the following upper bounds.Then we use the results of Theorem 1 and Theorem 2 to obtain the upper bound of(?)λsymi f×symj g(n)by using the partial sum formula and the method of reference[4].Furthermore,we use decomposition of generating functions and Dirichlet convolution to obtain the upper bound of RankinSelberg L-functions.In Chapter 4,based on Kühleitner and Nowak’s omega Theorem,we establish Ωresults of the sum(?)λf×symjg(n)and(?)λf(n)λg(nj)for j=1,2.In Chapter 5,inspired by Hua,we establish the mean value estimate of pairwise maxima of |λsymif×symjg(n)|2φand |λf(ni)λg(nj)|2φ.Let Sr*be the set of all normalized primitive Maass cusp forms of Laplace eigenvalue λ=1/4+r2 for the full modular group.Since the Ramanujan conjecture for Fourier coefficients of Maass cusp form hasn’t been proved yet,the proofs of Theorem 1 and Theorem 3 are not avaliable at all.In Chapter 6,we use the upper bound results of(?)|kf(n)|2δ and(?)|λsym2f(n)|,and the asymptotic formulas of(?)|λf(n)|2|λg(n)|2 and(?)|λf(n)|2|λg(n2)|2 to obtain the upper bounds of coefficients of Rankin-Selberg L-functions through Dirichlet convolution.