Dirichlet series associated to Modular forms

Abstract

Waring's problem, Goldbach's conjecture, Dirichlet divisor problem and the order of Dirichlet series in critical strip are good examples for the phenomena in Number theory, 'for a problem reducing in the ultimate analysis to estimating an exponential sum'. This thesis is concerned with functional equations for Dirichlet series and their twists by additive characters associated to arithmetic modular forms, for congruence subgroups of the full modular group SL(2, Z). Transformation formulae for exponential sums involving Fourier Coefficients of cusp forms are obtained; To show that "all those applications of transformation formula, obtained by M. Jutila, for 'Dirichlet series associated to cusp forms, for the full modular group' are valid in the case of Dirichlet series coming from arithmetic cusp forms for congruence subgroups of SL(2, Z)", two sample applications of the transformation formula are given as examples. Some theorems, Lemmas and corollary are explained and proved.