两个整数平方和序列中傅里叶系数的变号问题

Sign-change of Fourier Coefficients in Sequence of Sum of Squares of Two Integers

设f(z)是全模群SL(2,Z)上权为偶数k的全纯Hecke特征型,λ(n)是f(z)的第n个标准化傅里叶系数.借助L-函数均值估计和L-函数亚凸界结果,利用复变函数积分的方法,主要研究了傅里叶系数λ(n)在稀疏序列即两个整数平方和序列{λ(n):n=c^(2)+d^(2),(c≥1,d≥1)}上的符号变化问题,得到了对于足够大的x和任意小的正数1ε,序列{λ(n):n=c^(2)+d^(2),(c≥1,d≥1)}在区间(x,2x]上变号远大于x^(16/57 -ε)次的定量结果.

Let f(z)be the holomorphic Hecke eigenform of even integral weight k for the full modular group SL(2,Z)andλ(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 sign-change of Fourier coefficientsλ(n)over sparse sequence{λ(n):n=c^(2)+d^(2),(c≥1,d≥1)}.This paper provides the quantitative result for sufficiently large x and arbitrarily small positive constantε,the sequence16{λ(n):n=c^(2)+d^(2),(c≥1,d≥1)}changes its signs much larger than x^(16/57 -ε) times in the interval(x,2x].