Sign Changes Of Fourier Coefficients In The Sequence Of Sums Of Squares Of Two Integers

Let f(z)be the normalized Hecke eigenform of even integral weight k for the space of holomorphic cusp forms Sk(F0(N))and λ(n)denotes its n-th normalized Fourier coefficient.In this paper,with the help of mean value estimates and subconvex bounds of the L-functions,we use the complex function integral method to study the problem of the sign change of Fourier coefficients λ(n)in the sequence of sum of squares of two integers,and obtain the following conclusions.Firstly,if N is an odd and square-free positive integer.f(z)∈ Sk(Γ0(N))is a normalized Hecke eigenform of even integral weight k and λ(n)denotes its n-th normalized Fourier coefficient.Then for arbitrarily small positive constant ε,there exist n1,n2 which can be written as sum of two squares and n1,n2<<k2+εN2+ε,such that λ(n1)λ(n2)<0.Secondly,if f(z)be a normalized Hecke eigenform of even integral weight k for the full modular group SL(2,Z)and λ(n)denotes its n-th normalized Fourier coefficient.The sequence{λ(c2+d2)}c,d≥1 changes its signs at least x13/50-ε times in the interval(x,2x]for sufficiently large x,where ε is arbitrarily small positive constant.