自守L-函数系数在不同稀疏整数序列中的Ω结果

The Omega result of coefficients of automorphic L-functions over different sparse sequences

Let k be a positive even integer, and H_^*k be the set of all normalized Hecke primitive eigencuspforms of weight k for Γ=SL₂(Z). The Fourier expansion of f∈H_^*k at the cusp ∞ is defined by f(z)=λ_f(n)n^(k-1)/2e^2πinz, where λ_f(n) is the eigenvalue of the (normalized) Hecke operator T_n. The Omega result for the summatory functionλ_f(nⁱ)λ_f(n^j) is investigated. Set E_1,2(f,x)=λ_f(nⁱ)λ_f(n^j)-c_j-1x, i=1, j=2,3, where c₁, c₂ is a suitable constant. Then it is proved that E_1,2(f,x)=Ω(x^5/12),E_1,3(f,x)=Ω(x^7/16).