On Sums of Hecke Series in Short Intervals
نویسنده
چکیده
The purpose of this paper is to obtain a bound for sums of Hecke series in short intervals which, as a by-product, gives a new bound for Hj( 1 2 ). We begin by stating briefly the necessary notation and some results involving the spectral theory of the non-Euclidean Laplacian. For a competent and extensive account of spectral theory the reader is referred to Y. Motohashi’s monograph [13]. Let {λj = κj+ 14} ∪ {0} be the eigenvalues (discrete spectrum) of the hyperbolic Laplacian ∆ = −y (
منابع مشابه
On Exponential Sums with Hecke Series at Central Points
(T ε ≤ K ≤ T ) are considered, where αj = |ρj(1)| (coshπκj) , and ρj(1) is the first Fourier coefficient of the Maass wave form corresponding to the eigenvalue λj = κ 2 j + 1 4 to which the Hecke series Hj(s) is attached. The problem is transformed to the estimation of a classical exponential sum involving the binary additive divisor problem. The analogous exponential sums with Hj( 1 2 ) or H j...
متن کاملPower Sums of Hecke Eigenvalues and Application
We sharpen some estimates of Rankin on power sums of Hecke eigenvalues, by using Kim & Shahidi’s recent results on higher order symmetric powers. As an application, we improve Kohnen, Lau & Shparlinski’s lower bound for the number of Hecke eigenvalues of same signs.
متن کاملThe True Order of the Riemann Zeta–function on the Critical Line
UDC 511.3+517.43+519.45 For the Riemann zeta-function on the critical line the terminal estimate have been proved, which had been conjectured by Lindelöf at the beginning of this Centure. The proof is based on the authors relations which connect the bilinear forms of the eigenvalues of the Hecke operators with sums of the Kloosterman sums. By the way, it is proved that for the Hecke series (whi...
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 کاملDynamic characterization and predictability analysis of wind speed and wind power time series in Spain wind farm
The renewable energy resources such as wind power have recently attracted more researchers’ attention. It is mainly due to the aggressive energy consumption, high pollution and cost of fossil fuels. In this era, the future fluctuations of these time series should be predicted to increase the reliability of the power network. In this paper, the dynamic characteristics and short-term predictabili...
متن کامل