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)=(?)?f(n)nk-1/2e(nz),where ?f(n)is the n-th eigenvalue of the Hecke operator Tn.According to Ramanujan-Petersson conjecture proved by Deligne[4],we have|?f(n)|?d(n).The Hecke L-function associated to f?Hk*is defined by L(f,s)=(?)?f(n)/ns=(?)(1-?f(p)p-s)-1(1-?f(p)p-s)-1,Re(s)>1,where local parameters ?f(p)and ?f(p)satisfy?f(p)=?f(p)+?f(p),|?f(p)|=|?f(p)|=?f(p)?f(p)=1.Hence for each prime number p there is a unique ?f(p)?[0,?]such that?f(p)=ei?f(p)+e-i?f(p)=2 cos ?f(p).Murty[31]first studied the sign changes of ?f(p)for p ?(x,x+x?],where 0 is a small positive number.He proved that at least one ?f(p)changes sign in this short interval,in particular,the number of sign changes of ?f(p)for p ?x is at least ax? for some a>0.Later,Meher,Shankhadhar and Viswanadham[29],Meher and Murty[28],and He[8]investigated the sign changes of the sequences{?f(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 ?f(n)?g(n)and ?f(n)?g(n2),respectively.In the first chapter,we study the behavior of the signs of ?f(nj)(j>3)and?f(ni)?g(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 {?f(n)?g(n)?h(n)}n?1.Theorem 1.1 Let f?Hk*be a nonzero cusp form,and ?f(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{?f(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 let?f(n)and ?g(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 {?f(ni)?g(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 ?f(n),?g(n)and ?h(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 {?f(n)?g(n)?h(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 ?f(n),Matomaki and Radziwill[26]proved that asymptotically half of the nonzero ?f(n)in the sequence{?f(n)}n?1 are positive and half of them are negative,namely,(?)#{u?x:?f(n)(?)0}/#{n?x:?f(n)?0}=1/2.In the second chapter,we consider the number of positive and negative coef-ficients in the sequences {?f(n)?g(n)}n?1 and {?f(n)?g(n)?h(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 ?f(n)and ?g(n)be the coefficients of L(f,s)and L(g,s),respectively.Then we have(?)#{n?x:?f(n)?g(n)(?)0}/#{n?x:?f(n)?g(n)?0}=1/2.Theorem 2.2 Let f?Hk*,g?Hk1*and h?Hk2*be three different nonzero cusp forms.Also let ?f(n),?g(n)and ?h(n)be the coefficients of L(f,s),L(g,s)and L(h,s),respectively.Then we have(?)#{n?x:?f(n)?g(n)?h(n)(?)0}/#{n?x:?f(n)?g(n)?h(n)?0}=1/2.In addition,by using the theory of uniform distribution modulo one,Kohnen,Lau and Wu[15]proved that when ?f(p)/2? is irrational,(?)#{j?x:?f(pj)(?)0}/x=1/2.Subsequently,Amri[1]proved that when 1,?f(p)/2?,?g(p)/2? are linearly indepen-dent over Q,(?)#{j?x:?f(pj)?g(pj)(?)0}/x=1/2.In the second chapter,we study the number of elements in the set {j?N*:?f(pj)?g(pj)?h(pj)(?)0}.Theorem 2.3 Let f?Hk*,g ? and h ? Hk2*be three different nonzero cusp forms.Also let ?f(n),?g(n)and ?h(n)be the coefficients of L(f,s),L(g,s)and L(h,s),respectively.Suppose that 1;?f(p)/2?,?g(p)/2?,?h(p)/2? are linearly independent over Q,then we have(?)#{j?x:?f(pj)?g(pj)?h(pj)(?)0}/x=1/2.