On Sign Changes Of Coefficients Of Automorphic L-functions

In analytic number theory,it is an interesting subject to study the signs of coefficients of automorphic L-functions.In this paper,we consider the following three problems:·the behavior on the signs of coefficients of automorphic L-functions in short intervals(x,x+xr];·the number of sign changes of coefficients of automorphic L-functions in long intervals[1,x];·the number of positive and negative coefficients of automorphic L-functions.For the first two problems,Meher and Murty[27]established a general theorem to study the sign changes of any sequence of real numbers.Hence we shall discuss these two problems together.Next,we introduce the main results of this paper.Let Hk*be the set of all normalized Hecke primitive eigencuspforms of even integral weight k≥2 for SL(2,Z).The Fourier expansion of f∈Hk*at the cusp ∞is f(z)=(?)λf(n)nk-1/2e(nz),where λf(n)is the n-th eigenvalue of the Hecke operator Tn.According to Ramanujan-Petersson conjecture proved by Deligne[4],we have|λf(n)|≤d(n).The Hecke L-function associated to f∈Hk*is defined by L(f,s)=(?)λf(n)/ns=(?)(1-αf(p)p-s)-1(1-βf(p)p-s)-1,Re(s)>1,where local parameters αf(p)and βf(p)satisfyλf(p)=αf(p)+βf(p),|αf(p)|=|βf(p)|=αf(p)βf(p)=1.Hence for each prime number p there is a unique θf(p)∈[0,π]such thatλf(p)=eiθf(p)+e-iθf(p)=2 cos θf(p).Murty[31]first studied the sign changes of λf(p)for p ∈(x,x+xθ],where 0 is a small positive number.He proved that at least one λf(p)changes sign in this short interval,in particular,the number of sign changes of λf(p)for p ≤x is at least axθ for some a>0.Later,Meher,Shankhadhar and Viswanadham[29],Meher and Murty[28],and He[8]investigated the sign changes of the sequences{λf(nj)}n≥1(j=1,2,3,4).In addition,for two different nonzero cusp forms f∈Hk*and g∈Hk1*,Kumari and Murty[18],Gun,Kumar and Paul[7]established lower bounds for the number of sign changes of λf(n)λg(n)and λf(n)λg(n2),respectively.In the first chapter,we study the behavior of the signs of λf(nj)(j>3)andλf(ni)λg(nj)(i≥ 1,j≥ 2)in short intervals,and derive some lower bounds for the number of sign changes in long intervals.Moreover,we consider simultaneous sign changes of coefficents of three different automorphic L-functions,that is,the sign changes of the sequence {λf(n)λg(n)λh(n)}n≥1.Theorem 1.1 Let f∈Hk*be a nonzero cusp form,and λf(n)be the coefficient of L(f,s).Then for j≥3 and any r with 1-84/42(j+1)2-37<r<1,the sequence{λf(nj)}n≥1 has at least one sign change for n∈(x,x+xr].Moreover,the number of sign changes for n ≤x is>>x1-r for sufficiently large x.Theorem 1.2 Let f ∈ and g∈Hk1*be two different nonzero cusp forms.Also letλf(n)and λg(n)be the coefficients of L(f,s)and L(g,s),respectively.Then for i≥1,j≥2 and any r with 1-42/21(i+1)2(j+1)2-29<r<1,the sequence {λf(ni)λg(nj)}n≥1 has at least one sign change for n ∈(x,x+xr].Moreover,the number of sign changes for n ≤x is>>x1-r for sufficiently large x.Remark Theorem 1.1 improves the results of Meher,Shankhadhar and Viswanad-ham for j=3,4,and Theorem 1.2 improves the result of Gun,Kumar and Paul.Theorem 1.3 Let f∈Hk*,g ∈ and h ∈ Hk2*be three different cusp forms.Also let λf(n),λg(n)and λh(n)be the coefficients of L(f,s),L(g,s)and L(h,s),respectively.Then for any r with 63*65<r<1,the sequence {λf(n)λg(n)λh(n)}n≥1 has at least one sign change for n ∈(x,x+xr].Moreover,the number of sign changes for n ≤x is>>1-r for sufficiently large x.By using Halasz’s theorem and the multiplicativity of λf(n),Matomaki and Radziwill[26]proved that asymptotically half of the nonzero λf(n)in the sequence{λf(n)}n≥1 are positive and half of them are negative,namely,(?)#{u≤x:λf(n)(?)0}/#{n≤x:λf(n)≠0}=1/2.In the second chapter,we consider the number of positive and negative coef-ficients in the sequences {λf(n)λg(n)}n≥1 and {λf(n)λg(n)λh(n)}n≥1,and establish two results similar to the one obtained by Matomaki and Radziwill.Theorem 2.1 Let f ∈Hk*and g∈Hk1*be two different nonzero cusp forms.Also let λf(n)and λg(n)be the coefficients of L(f,s)and L(g,s),respectively.Then we have(?)#{n≤x:λf(n)λg(n)(?)0}/#{n≤x:λf(n)λg(n)≠0}=1/2.Theorem 2.2 Let f∈Hk*,g∈Hk1*and h∈Hk2*be three different nonzero cusp forms.Also let λf(n),λg(n)and λh(n)be the coefficients of L(f,s),L(g,s)and L(h,s),respectively.Then we have(?)#{n≤x:λf(n)λg(n)λh(n)(?)0}/#{n≤x:λf(n)λg(n)λh(n)≠0}=1/2.In addition,by using the theory of uniform distribution modulo one,Kohnen,Lau and Wu[15]proved that when θf(p)/2π is irrational,(?)#{j≤x:λf(pj)(?)0}/x=1/2.Subsequently,Amri[1]proved that when 1,θf(p)/2π,θg(p)/2π are linearly indepen-dent over Q,(?)#{j≤x:λf(pj)λg(pj)(?)0}/x=1/2.In the second chapter,we study the number of elements in the set {j∈N*:λf(pj)λg(pj)λh(pj)(?)0}.Theorem 2.3 Let f∈Hk*,g ∈ and h ∈ Hk2*be three different nonzero cusp forms.Also let λf(n),λg(n)and λh(n)be the coefficients of L(f,s),L(g,s)and L(h,s),respectively.Suppose that 1;θf(p)/2π,θg(p)/2π,θh(p)/2π are linearly independent over Q,then we have(?)#{j≤x:λf(pj)λg(pj)λh(pj)(?)0}/x=1/2.