The Spectral Mean Square of Hecke L-functions on the Critical Line
نویسندگان
چکیده
The Hecke L-function Hj(s) attached to the jth Maass form for the full modular group is estimated in the mean square over a spectral interval for s = 1 2 + it. As a corollary, we obtain the estimate Hj( 1 2 + it) ¿ t1/3+ε for t À κ j , where 1/4 + κj is the respective jth eigenvalue of the hyperbolic Laplacian. This extends a result due to T. Meurman.
منابع مشابه
On Hecke L-functions attached to half-integral weight modular forms
We would like to recall that in the case of Hecke eigenforms on Γ1 non-vanishing results for their Hecke L-functions at an arbitrary point s0 in the critical strip (not on the critical line) have been proved in [4] (cf. also [7]), using holomorphic kernel functions. This method was carried over to the case of half-integral weight in [8], for arbitrary level. However, in this approach for given ...
متن کاملAn Explicit Formula for Hecke L-functions
In this paper an explicit formula is given for a sequence of numbers. The positivity of this sequence of numbers implies that zeros in the critical strip of the Euler product of Hecke polynomials, which are associated with the space of cusp forms of weight k for Hecke congruence subgroups, lie on the critical line.
متن کاملOn the mean values of L-functions in orthogonal and symplectic families
Hybrid Euler-Hadamard products have previously been studied for the Riemann zeta function on its critical line and for Dirichlet Lfunctions in the context of the calculation of moments and connections with Random Matrix Theory. According to the Katz-Sarnak classification, these are believed to represent families of L-function with unitary symmetry. We here extend the formalism to families with ...
متن کاملar X iv : m at h / 05 05 39 9 v 1 [ m at h . N T ] 1 9 M ay 2 00 5 Uniform bound for Hecke L - functions
Most of arithmetically significant Dirichlet series such as the Riemann zeta-function ζ(s), Dirichlet L-functions, and Hecke L-functions associated with various cusp forms satisfy Riemannian functional equations connecting values at s = σ + it and 1 − s of respective functions. Essentially best possible estimates for these functions near the line σ = 1 and σ = 0 can usually be deduced from the ...
متن کاملOn Some Conjectures and Results for the Riemann Zeta-function and Hecke Series
We investigate the pointwise and mean square order of the function Z2(s), where Zk(s) = ∫∞ 1 |ζ(12 + ix)|2kx−s dx, k ∈ N. Three conjectures involving Z2(s) and certain exponential sums of Hecke series in short intervals are formulated, which have important consequences in zeta-function theory. A new order result for Z2(s) is obtained, and the function Zk(s) is discussed. 1. Spectral theory and ...
متن کامل