A Mean Value Theorem For Automorphic L-functions

Generally speaking, an L-function is a type of generating function formed out of local data with either an arithmetic-geometric object, such as an elliptic curve defined over a number field, or an automorphic form. According to conjectures in the Langlands program, any "most general" L-function should in fact factorize into a product of L-functions of automorphic cuspidal representations of GLm/Q. Thus these automorphic L-functions of GLm/Q are building blocks for all "most general" L-functions, and hence deserves deep investigation.In this thesis, we study automorphic L-functions attached to holomorphic cusp forms of GL2/Q.We begin our discussion with the case of GL1. Let ζ(s) with s = σ + it be the Riemann zeta-function. Hardy and Littlewood investigated the integral mean squareand proved that, as T →∞This means that the average value of is logt. Assuming the Riemann Hypothesis and denoting by γ < γ+ successive ordinates of zeros of ζ(s), Conrey and Ghosh defined the discrete mean squareand established the formulawhere N(T) is the number of non-trivial zeros p = 1/2 + iγ with 0 < γ < T. This shows that the average of at a critical point is logt. Note thatis larger than 1. Also, we may rewrite the formulae (0.1) and (0.2)In this thesis, we investigate this phenomenon for the automorphic L-functions L(f,s) attached to a holomorphic cusp forms f of SL2(Z). DefineAssume the Generalized Riemann Hypothesis for L(f,s) that all the non-trivial zeros of L(f, s) lie on the critical line σ=1/2. Let γ < γ+ be successive ordinates ofzeros of L(f, s), andwhere N(f, T) is the number of non-trivial zeros p = 1/2+iγ of L(f, s) with 0 < γ < T.We remark thatthis can be compared with the classical formulafor the Riemann zeta-function.Our main result is the followingTheorem 0.1. Let L(f, s) be the automorphic L-functions attached to holomorphic cusp forms f of SL2(Z), and assume the Generalized Riemann Hypothesis for L(f,s). Thenand r(n) certain coefficients depending on f.Our L(f, s) is an L-function of order 2, while ζ(s) is of order 1. From (0.8), it is clearly seen that L(f.s) is essentially more difficult than ζ(s). We prove in Lemma 7.1 thatUnfortunately, however, current techniques do not seem capable of deriving an asymptotic formula for H(T) via (0.8). We therefore give in §7 heuristic arguments, which suggest thatand if H(T) is of order T log2 T, then an asymptotic formula H(T) ~ c1T log2 T exists. This leads further toConjecture 0.2. There exists a constant c > 0, such thatThe constant c depends at most on the weight k of f.