On the fourth power moment of Fourier coefficients of cusp form.

Let a(n) be the Fourier coefficients of a holomorphic cusp form of weight κ = 2n ≥ 12 for the full modular group and A(x) = Ʃn≤xa(n). In this paper, we establish an asymptotic formula of the fourth power moment of A(x) and prove that ∫1TA4(x)dx=364κπ4s4;2(a˜)T2κ+O(T2κ−δ4+ε)<inline-graphic></inline-graphic> with δ4 = 1/8, which improves the previous result. [ABSTRACT FROM AUTHOR]