On the moments of Hecke series at central points

Abstract

We prove, in standard notation from spectral theory, the following asymptotic formulas: $$\sum_{\kappa_{j}\leqslant K} \alpha_{j}H^3_j(\frac{1}{2}) = K^2 P_3 (\log K) + O(K^{5/4} \log^{37/4}K)$$ and $$\sum_{\kappa_{j}\leqslant K} \alpha_{j}H^4_j(\frac{1}{2}) = K^2 P_6 (\log K) + O(K^{3/2} \log^{25/2}K),$$ where $P_3 (x)$ and $P_6 (x)$ are polynomials of degree three and six, whose coefficients may be explicitly evaluated.