Double square moments and subconvexity bounds for Rankin-Selberg L-functions of holomorphic cusp forms

Let f and g be holomorphic cusp forms of weights k1 and k2 for the congruence subgroups TO(N1)and Γ0(N2),respectively.In this paper the square moment of the Rankin-Selberg L-function for f and g in the aspect of both weights in short intervals is bounded,when k1^ε <<k^2<<k1^1-ε.These bounds are the mean Lindelof hypothesis in one case and subconvexity bounds on average in other cases.These square moment estimates also imply subconvexity bounds for individual L(1/2+it,f×g) for all g when f is chosen outside a small exceptional set.In the best case scenario the subconvexity bound obtained reaches the Weyl-type bound proved by Lau et al.(2006) in both the k1 and k2 aspects.

Oscillations of coefficients of symmetric square L-functions over primes

Let L（s, sym2f） be the symmetric-square L-function associated to a primitive holomorphic cusp form f for SL（2, Z）, with tf（n, 1） denoting the nthcoefficient of the Dirichlet series for it. It is proved that, forN≥2and anyα∈ there exists an effective positive constant c such that ∑n≤N∧（n）tf（n,1）e（nα）〈〈N exp （-c√logN,where ∧（n） is the von Mangoldt function, and the implied constant only depends on f. We also study the analogue of Vinogradov＇s three primes theorem associated to the coefficients of Rankin-Selberg L-functions.

Integral mean square estimation for the error term related to n xλ2(n2)

Let λf(n) be the n-th normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) ∈Sk(Γ).We establish that, for any ε > 0,1/Xintegral from n=1 to x|sum λ~2f^((n^2)) from n≤x to - c_2x|2dx ?ε X154/101+ε,which improves previous results.

A Criterion for the Generalized Riemann Hypothesis

In this paper, we study the automorphic L-functions attached to the classical automorphic forms on GL（2）, i.e. holomorphic cusp form. And we also give a criterion for the Generalized Riemann Hypothesis （GRH） for the above L-functions.