The Average Behavior Of Fourier Coefficients Of Cusp Forms Over Sequence Of Sums Of Two Squares

The Fourier coefficients of automorphie forms are interesting and important ob-jects due to their arithmetic significance. There are many problems concerning the Fourier coefficients, such as the famous Ramanujan-Peterson conjecture. In the mean-time, as an important problem in number theory, the average behavior of the Fourier coefficients has been investigated by many number theorists. The aim of this thesis is to study the average behavior of Fourier coefficients of holomorphic cusp forms over sequence of sums of two squares.More precisely, letλ(n) be the n-th normalized Fourier coefficient of holomorphic Hecke eigencuspform f(z)∈SL(2,Z) of even integral weight and S denote the set of numbers representable as a sum of two squares. In this thesis, we consider three types of sums involving Fourier coefficients of cusp forms over S, i.e.In 1908, Landau gave an estimate of the size of numbers representable as a sum of two squares. To solve the problem, he introduced three L-functions, and applied the classical methods of prime number theory to a function with an algebraic singu-larity. We partially follow Landau's idea for solving classical problems to establish the asymptotic formulae for the sums. By the works of Lu [21], [23], [24] and using the properties of symmetric power L-functions, Rankin-Selberg symmetric power L-functions and their twisted L-functions, we are able to obtain the following results for Sk(x),k= 1,2,…8. where c is a constant. where P3(x) is a polynomial of degree 3. where P8(x) is a polynomial of degree 8. where P20(x) is a polynomial of degree 20.By the works of Lu [22] and using the properties of symmetric power L-functions and their twisted L-functions, we are able to obtain the following results for S(k)(x), k≥ 1:under Serre conjecture, we have for k≥1, where the cases for k= 1,2,3,4 are unconditional results.By the works of Lao and Sankaranarayanan [19] and using the properties of Rankin-Selberg symmetric power L-functions and their twisted L-functions, we are able to obtain the following results for S2(k)(x), k= 1,2,3,4; where ci is a constant.