Sums Of Fourier Coefficients Of Powers Of The△~K(z)-Function

For an integer r, define the number pr(n) byIt is an important problem to study pr(n). For example, when r=1, we recover the classical partition generating functionWhen r=1, we recover the identity of EulerWhen r=3, we recover the identity of JacobiIn a series of papers [18]-[21], Newman studied the function pr(n) and compute pr(n) for small n. Another natural question is about hoe large(as a function of r and n)thepr(n) are. Deligne [3] showed that at least when r is even.In the case of r=24, it implies Ramanujan’s famous conjecture that if is the Fourier expansion of weight12cusp form△(z), thenWhen r=24k, Rouse [22] computed explicit bounds for the coefficients pr(n), when r>0and is a multiple of24, note is the representation of△k(z) as a sum of Hecke eige ThenThis paper aim to compute∑prj(n). Especially, we computewhen j=1,2,3,4,5,6. This paper is mainly composed of three parts. The first part introduces systematically the background of the subject and gives the result:Theorem1.1is any cusp form of weight m and write where each fi(z) is a Hecke eigenform and λi(n) denotes its nth normalized coefficient. Let When dim Sm=2, then for any ε>0,we have where P(x) is a polynomial of degree4.Theorem1.2when k=2,we have where P(x) is a polynomial of degree4.Theorem1.3Let is any cusp form of weight m and write where each fi(z) is a Hecke eigenform and A;(n) denotes its nth normalized coefficient. Let When dim Sm=3, then for any ε>0, we have Theorem1.4Let n=1and write Cjfi, where each fi(z) is a Hecke eigenform and λj(n) denotes its nth normalized coefficient. Let Then for any ε>0, we haveThe second part introduces the preliminary knowledge useful to prove the theorems, including the definition of Rankin-Selberg L-function, of j-th L-function, and the archimedean local factor.The third part is the proof of these theorems. The proof in this paper applies many methods and skills of analytic number theory, and uses Perron’s formula, Cauchy’s residue theorem.