| Modular forms is an interesting and important topic in modern analytic number theory,and it has attracted lots of attentions in modern literatures.The celebrated Langlands program proposed by R.P.Langlands attempts to build the bridges of analysis,algebra and geometry,and this vast perspective has generated wide and deep influence to the whole spectrum of mathematics.One of the most important focus in the Langlands program is the theory of modular forms,and this program has also governed the development of modern analytic number theory for more than fifty years.The analytic theory of modular forms is one of the most important branches of modular forms,and the analytic properties of Fourier coefficients of modular forms has been studied in various aspects by a number of famous mathematicians.The sign changes and non-vanishing of Fourier coefficients attached to modular forms recently have been received considerable attentions in the literatures,and there are a huge number of achievements in this direction.This topic has gained increasing concern in recent times.In this thesis,we mainly consider the sign changes and non-vanishing of Fourier coefficients attached to modular forms in three aspects:·The sign changes of Fourier coefficients of primitive holomorphic cusp forms;· The sign changes of Fourier coefficients of Hecke-Maass cusp forms;·The non-vanishing of Fourier coefficients of holomorphic cusp forms.Firstly,we consider the sign changes of normalized Fourier coefficients of primitive holomorphic cusp forms.Let f∈Hk*be a normalized primitive Hecke cusp form of integral weight k for the full modular group Γ=SL(2,Z),and denote byλf(n)the n-th normalized Fourier coefficient of f,it can be proved quite easily that the sequence {λf(n)}n∈N has infinitely many sign changes by using a classical theorem of Landau and certain analytic properties of associated L-functions(see e.g.[37]).For j≥ 1,let λsymj f(n)be the n-th normalized coefficient of the Dirichlet expansion of j-th symmetric power L-function L(symj f,s)attached to f,and denote by nsymj f be the least integer n such that λsymj f(n)<0.The upper bound of nsymjf has been considered by some authors for j=1,2,3,4(see e.g.[28,40,42,52,64,65]).Let π=?pπp be a self-contragredient irreducible unitary cuspidal representation for GLm(AQ).Denote by λπ(n)the n-th normalized coefficient of the standard global L-function L(s,π)attached to π.Here we normalize λπ(n)with λπ(1)=1.Let nπ be the least integer n such that λπ(n)<0.Let Qπ be the analytic conductor of L-function L(s,π).In[59],Liu,Qu and Wu proved nπ?πQπ1+?,which refined the previous result of Qu[81]with the exponent m/2.In this thesis,the author consider the cases j≥ 5 and establish the following result.Theorem 0.1.Let f∈Hk*be a Hecke eigenform.And let nsymj f denote the least integer n such that λsymj f(n)<0.Then for j≥ 5 any fixed positive integer,we have nsymj f?q(symj f)1/2+?for any ε>0,where the implied constant is absolute and q(symjf)is defined byA number of author considered the sign changes of Fourier coefficients of holomorphic cusp forms at the subsequences.In the early 1980s,R.Murty[74]considered the sign changes of the sequences of Fourier coefficients of holomorphic cusp forms at primes numbers.For more details on this topic,we refer the interested readers to[1,39,46,68,91,92].In particular,Vaishya[91,92]proved the quantitative results of normalized Fourier coefficients of Hecke newforms over a certain integral binary quadratic form.Inspired by the results of Vaishya[91,92]and the recent works of Newton and Thorne[75,76],along with the nice analytic properties of the associated L-functions,the author establish the following theorem.Theorem 0.2.Let j≥1 be any fixed positive integer.Let f∈Hk*be a normalized Hecke eigenform.Let Q(X)be a primitive integral positive definite binary quadratic reduced form with fixed discriminant D<0.In addition,we also assume that the class number h(D)=1.Then the sequence {λ,(nj)}(?)has at least?x1-δ sign changes in the interval(x,2x]with 1-210/210(j+1)2-103<δ<1 for sufficiently large x.In particular,the sequence {λf(nj)}(?)has infinitely many sign changes.In a similar manner,we also establish the result concerning the sign changes of Fourier coefficients associated to two distinct holomorphic cusp forms over the integral binary quadratic form.Theorem 0.3.Let i≥ 1 and j≥ 1 be two positive integers.Let f∈Hk1*and g ∈ Hk2*be two distinct normalized Hecke eigenforms.Let Q(X)be a primitive integral positive definite binary quadratic reduced form with fixed discriminant D<0.In addition,we also assume that the class number h(D)=1.Then the sequence {λf(ni)λg(nj)}(?)has at least ? x1-δ sign changes in the interval(x,2x]with 1-105/105(i+1)2(j+1)2-83<δ<1 for sufficiently large x.In particular,the sequence {λf(ni)λg(nj)}(?)has infinitely many sign changes.On the other hand,some authors considered the sign change problems of Fourier coefficients associated with two distinct holomorphic cusp forms.For more details,we refer to[16,41,44].In this direction,the author proved some results concerning the sign changes of coefficients of Dirichlet expansions of triple product L-functions associated to one or two distinct holomorphic cusp forms.We have the following result.Theorem 0.4.Let f ∈ Hk1*and g∈Hk2*be two distinct Hecke eigenforms,and denote by λf×f×f(n),λf×f×g(n)the n-th normalized coefficients of the triple product L-functions L(f × f ×f,s),L(f × f × g,s)attached to f and f,g,respectively.Then(ⅰ)For δ>2903/3008,the sequence {λf×f×f(n)}n∈N changes its signs at least ?x1-δtimes in the interval(x,2x]for sufficiently large x.In particular,the sequence{λf×f×f(n)}n∈N has infinitely many sign changes.(ⅱ)For η>1223/1265,the sequence {λf×f×g(n)}n∈N changes its signs at least ?x1-ηtimes in the interval(x,2x]for sufficiently large x.In particular,the sequence{λf×f×g(n)}n∈N has infinitely many sign changes.A number of authors considered the number of sign changes of Fourier coefficients associated to one or two cusp forms over certain sparse sequences(see e.g.[17,44,68]).Later,Lao and Luo[46]refined and generalized the previous results for certain sparse sequences and provided the quantitative results in this direction.In particular,they proved that for f∈Hk1*,g∈Hk2*and h ∈ Hk3*being three distinct Hecke eigenforms,the sequence {λf(n)λg(n)λg(n)}n∈N has at least?x1-r sign changes with 63/65<r<1 for n ≤x.Let f ∈ Hk1*,g ∈ Hk2*and h∈Hk3*be three distinct Hecke eigenforms,and denote by λf×g×h(n)the n-th coefficient of the Dirichlet expansion of the triple product L-function L(f × g×h,s)attached to f,g and h.By following essentially the same argument as that of Lao and Luo,the author establish the following result.Theorem 0.5.Let f ∈Hk1*,g∈Hk2*and h∈Hk3*be three distinct Hecke eigenforms.Then for any δ with 63/65<δ<1,the sequence {λf×g×h(n)} has at least one sign change for(x,xδ]for sufficiently large x.Furthermore,the numberof sign changes of the same sequence for n ≤x is?x1-δ.Secondly,we consider the sign changes of Fourier coefficients of Hecke-Maass form.Let Sv be the set of normalized primitive Hecke-Maass cusp forms of weight zero with Laplacian eigenvalues λ=1/4+v2 for the full modular groupΓ=SL(2,Z).Let φ∈ Sv be a Hecke-Maass cusp form,and denote its n-th normalized Fourier coefficient by λφ(n).As is point out in the last part of[37],one can prove that the sequence {λφ(n)n∈N has infinitely many sign changes by using the similar method.In[80],Qu considered the first sign change of the sequence {λφ(n)}n∈N and gave a upper bound for the first sign change,and Lau et al.[55]recently computed the numerical value of the result of Qu.Inspired by the results of sign changes of Fourier coefficients of holomorphic cusp forms over sparse sequences(see e.g.[1,44]),the author establish the following results by using the similar approach in these literature.Theorem 0.6.Let φ∈ Sv be a Hecke-Maass cusp form,and denote by λφ(n)the nth normalized Fourier coefficient of φ.Assuming the Ramanujan conjecture for φ,we have the following:(ⅰ)For any δ>3/5-1/560,the sequence {λφ(n)}n∈N has at least? x1-δ sign changes in the interval(x,2x]for sufficiently large x,where ε>0 is an arbitrarily small positive number.In particular,there are infinitely many sign changes of the sequence {λφ(n)}n∈N;(ⅱ)For any η>535/703,the sequence {λφ(n2)}n∈N has at least ? x1-η sign changes in the interval(x,2x]for sufficiently large x for any ε>0.In particular,there are infinitely many sign changes of the sequence {λφ(n2)}n∈N.Theorem 0.7.Let φ∈Sv be a Hecke-Maass cusp form,and denote by λφ(n)the nth normalized Fourieb coefficient of φ.For δ*>23/32,then the sequence{λφ(c2+d2)}c,d≥1 has at least?x1-δ*sign changes in the interval(x,2x]for sufficiently large x,where ε>0 is an arbitrarily small positive number.Let f and g be two distinct normalized Hecke-Maass cusp forms,and denote its normalized Fourier coefficients by λf(n)and λg(n),respectively.Let nf,g be the least positive integer n such that λf(n)λg(n)<0.In[45],Kumari and Sengupta for the first time proved two results concerning the simultaneous sign changes of the sequence {λf(n)λg(n)}n∈N.Based on the ideas of[45]and some well-known results(see e.g.[72]),along with some simple analytic techniques,the author refine the results of Kumari and Sengupta and prove the following results.Theorem 0.8.Let f and g be two distinct normalized Hecke-Maass cusp forms with Fourier coefficients λf(n)and λg(n),respectively.Then the sequence{λf(n)λg(n)n∈N changes its signs at least?x1/16-δ times in the interval(x,2x]for x sufficiently large,where δ∈(0,1/16)be any fixed sufficiently small number.Theorem 0.9.Let f ∈ Su and g ∈ Sv be two distinct normalized Hecke-Maass cusp forms.Then where c>0 is some suitable absolute constant.Here q(sym2f)=3(3+u)2 and q{sym2g)=3(3+v)2 are the analytic conductors of the symmetric square L-functions associated with f and g,respectively.In the last part,we consider the non-vanishing problems of Fourier coefficients of holomorphic cusp forms.Let f∈Hk*be a Hecke eigenform.Let λsymj f(n)denote the n-th normalized coefficient of the Dirichlet expansion of the j-th symmetric power L-function L(symj f,s)attached to f.Let where j≥1 is any fixed positive integer.In the literatures,the authors only considered the cases j=1,2(see e.g.[38,49,50,93]).Let π=?pπp be a selfcontragredient irreducible unitary cuspidal representation for GLm(AQ).Denote by λπ(n)the n-th normalized coefficient of the standard global L-function L(s,π)attached to π.For π corresponds to the symmetric power L-function L(symj f,s),by a result of Lau,Liu and Wu[54],one has the following Nsymj f±(x)?f x1/j+1 for all j≥1.In this thesis,we establish the asymptotic formulae for Nsymj f*(x)and Nsymj f±(x),which significantly improves and generalizes the previous results in this direction.Theorem 0.10.Let f∈Hk*be a Hecke eigenform.Then(ⅰ)We have Nsymj f*(x)=ρsymj fx{1+Of(1/(log x)δ)}(j≥1)as x→∞,where 0<δ<1/2,and Here P denotes the set prime numbers,and δsymj f(n)is the characteristic function of n such that λsymj f(n)≠0.(ⅱ)We have·For j≥1 a odd fixed integer,then Nsymj f±(x)=1/2ρsymj fx{1+Of(1/(log x)K)}as x→∞;·For j≥2 a even fixed integer,then Nsymj f±(x)=1/2ρsymj fx{1+Of(1/(log x)βjK)},whereβj=j/j+1+1/π tan(π/j+1).Here ρsymj f is defined by(0.2),and K=0.32867...=-cos φ0 and φ0 is the unique root in(0,π)of the equation sin φ-φ cos φ=1/2π.DefineTheorem 0.11.Let f∈ Hk1*,g∈Hk2*and h E Hk3*be three distinct Hecke eigenforms.Then there exists suitable constant x0(f,g,h)>0 such that Nf,g,h±(x)?f,g,h x holds for all x≥ x0(f,g,h).Based on the work of Kowalski-Robert-Wu[43],we are able to establish the following theorem.Theorem 0.12.Let f ∈ Hk1*,g ∈ Hk2*and h∈Hk3*be three distinct Hecke eigenforms.Then there exists a set S of primes of with natural density one such that for any p ∈ S,we haveλf×g×h(p)≠0.By a general result of Gun,Kumar and Paul[17]concerning non-vanishing theorem for certain multiplicative function.We have the following theorem.Theorem 0.13.Let f∈Hk1*,g ∈ Hk2*and h∈Hk3*be three distinct Hecke eigenforms.Then(ⅰ)For any ε>0,x≥x0(f,g,h,ε)and y≥x7/17+ε,we have#{x<n ≤x+y | n square-free and λf×g×h(n)≠0}?f,g,h,ε y.(ⅱ)For any ε>0,x≥x0(f,g,h,ε),y≥ x17/38+ε and 1 ≤a ≤q ≤xε,(a,q)=1,we have#{x<n≤x+y | n square-free,n≡a(modq)and λf×g×h(n)≠0}?f,g,h,ε y/q. |
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