Sign Changes Of The Coefficients Of Triple Product L-function

Automorphic L-functions are important objects in analytic number theory,and triple product L-functions are a class of automorphic L-functions with broad applications.In this paper,our main objective is to study the sign changes of coefficients of triple product L-functions using analytic techniques.We consider the holomorphic cusp forms or Maass cusp forms for the full modular group.Denote by f∈Hk*the normalized primitive holomorphic cusp form of weight k of trivial nebentypus for the full modular group,where k≥ 2 is an even integer,λf(n)∈R is eigenvalues of Hecke operator Tn.A series of articles in the literature are devoted to investigations in the number of sign changes of Fourier coefficients(see[1],[10],[20],[23],[27],[30],[31]and[42]).In this paper,we firstly decompose the corresponding Dirichlet series into a product of symmetric L-functions and a simpler Dirichlet series.Then we use Perron’s formula and Cauchy’s residue theorem to obtain the mean value of Fourier coefficients of triple product L-function L(f × f × f,s).At last,we can use Lemma 2.7 to get the number of sign changes of λf×f×f(n)in natural number set and{n|n=c2+d2,(c,d)∈N2}.Furthermore,under the Generalized Lindelof Hypothesis,we improve the lower bounds of number of sign changes of λf×f×f(n)in the above two sets.Let g ∈Mr*be a normalized primitive Maass cusp form of Laplace eigenvalue λ=1/4+r2.Many scholars studied the mean value of Fourier coefficients of triple product L-function.Since the bound of λg(n)<<nε is not avaliable yet,Perron’s formula can not be applied in its full strength.We will take advantage of Landau’s Lemma and Dirichlet’s convolution to obtain the mean value of coefficients of triple product L-functions L(f ×f × g,s).Then we will use Lemma 2.7 to obtain quantitative results for the sign changes of λf×f×g(n)in natural number set and {n|n=c2+d2,(c,d)∈ N2}.