PETERSSON'S CONJECTURE FOR CUSP FORMS OF WEIGHT ZERO AND LINNIK'S CONJECTURE. SUMS OF KLOOSTERMAN SUMS

Abstract

This paper begins a series of articles whose purpose is to obtain estimates for the Fourier coefficients of the eigenfunctions of the discrete spectrum of the Laplace operator on the Lobachevsky plane which are automorphic for the modular group. Identities are obtained which express sums of Kloosterman sums in terms of bilinear combinations of the Fourier coefficients of the eigenfunctions of the Laplace operator. The mean value of Kloosterman sums is also estimated, and an asymptotic formula is given for the mean value of the square of the modulus of the Fourier coefficients of the eigenfunctions of the discrete spectrum. Bibliography: 24 titles.