Estimates On Analytic Density Of Hecke Eigenvalues Of Cusp Forms

For the holomorphic cusp modular form,the coefficients of its Fourier expansion are closely related to the eigenvalues of the Hecke operator,therefor,the sign and size of Hecke eigenvalues has always been a hot topic in analytic number theory.Let holomorphic cusp form of weight k?2N+ and level N?N+ be f(z)=(?)?f(n)n(k-1)/2e2?inz,in 2009,Y.-K.Lau and Jie Wu constructed a special set of numbers and proved[4]that there exists a constant x0(f)>0 such that for x?x0(f),the number of positive or negative Hecke eigenvalues(?).That is,in the range of integers,the set {nl?f(n)(?)0} has positive density,and the coeffcient of proportionality is related to the function f.Recently,Chiriac compared the Hecke eigenvalues of two different holomorphic cusp modular forms f and g,he used the method in[4]and combined the results of the Sato-Tate conjecture and obtained similar estimation.The estimation says that there is a certain proportion of integers n such that ?f(n)<?g(n),where ?g(n)is the nth Hecke eigenvalue of the modular form g.In fact,the above estimation of Hecke eigenvalues is equivalent to natural density estimation.In addition,analytic density(also called Dirichlet density)is another density estimate method for the set of prime numbers,which has certain advantages over natural density when estimating density of sets of prime numbers.In 201 7,Chiriac raised the question in[1]:Are there two different holomorphic cusp modular forms f and g,such that the inequality ?f(p)<?g(p)holds for every prime number p?At the same time,he proved that for two different holomorphic cusp modular forms f and g,the set of prime numbers{p|?f(p)<?g(p)}has analytic density at least 1/16,which means that the situation in the above question does not exist.In addition,the analytic density of the set{p|?f(p)<?g(p)}is also at least 1/16.Therefore,the analytic density of the set{p|?f(p)<?g(p)}can not be greater than 7/8.Besides,Chiriac used the same method to prove that when f and g do not have complex multiplication,which is equivalent to say that when there is not a Dirichlet character ?,such that f=?(?)g,the set{p|?f2(p)<?g2(p)}has analytic density at least 1/16,the result is equivalent to comparing the absolute value of Hecke eigenvalues.By applying symmetric power L-function,Rankin-Selberg L-function and the properties of cusp automorphism,we use Chiriac's main method in[1]to prove that for two different holomorphic cusp modular forms f and g,after giving some restrictions,the set of prime numbers{p|?f(p)+m?g(p)+n<0}has analytic density at least 1+m2-2n(1+|m|)/4(2+2|m|-n)(1+|m|).Besides,when m ? 0,the set{p|?f2(p)+m?g2(p)+n<0}has analytic density at least-2m2-6m-2-3n-3mn/4(1+m)(-n),the set{p|?f2(p)+m?g2(p)+n<0}has analytic density at least 2m2-2m-2-3n+mn/4(-4m-n)(-m+1).