On The Estimation Of Nonlinear Twists Of The Liouville Function

In analytic number theory,the problems concerning nonlinear exponential twisting arithmetic functions arise naturally in investigating equi-distribution the-ory,zero-distribution of L-functions and so on.We usually consider the general nonlinear exponential sum of the form Here,n～X means X ≤ n ≤ 2X and e(z)=e2πriz.When β=1/2,the sum S(X,a)was studied by Vinogradov[12]for the von Mangoldt function an=∧(n).For an=∧(n)and an=μ(n)(μ is the Mobius function)the sums S(X,α)were studied by Iwaniec,Luo and Sarnak[3],and they showed these sums are intimately related to L-functions of GL2.If f is a holomorphic cusp form of even weight on the upper half plane,they also proved that a good upper bound of(X,a)implies a quasi Riemann hypothesis for L(s,f)[3].If β is variable and an are the Fourier coefficients of automorphic forms.Ren and Ye[7]received that:Let 0<β<1,0≠α∈R.(i)For β≠1/2 and all α≠0,we have(ii)For |α| ≤1 or |α|≥X1/2,we have(iii)For 1<|α|a<X1/2,if there exists a positive integer q satisfying ||α|-2(?)q|<X-1/2(such a q is unique),then where c0 is a non-zero constant defined by if such a q does not exist,then it holds without the leading term.(iv)In particular,if α=±2(?)q with integer 1 ≤ q<X/4,then it becomes where c0 can be written as c0=±(-1)k+12/3(23/4-1)(1+i).For prime powers pk,the Liouville function λ(n)=∑d2|n μ(n/d2),λ(pk)=(-1)k.Throughout this paper,we consider the sum S(X,α)=∑n～X λ(n)e(αnβ).(1.1)As noticed in Murty and Sankaranarayanan[4],it is amusing to point out that the hypothesis that for some θ<1,∑ n≤X λ(n)=O(Xθ)implies the quasi Riemann hypothesis.It should be pointed out that the Riemann hypothesis is equivalent to the view that the above est.imate holds for every θ>1/2.Also,Iwaniec,Luo and Sarnak considered the sums with an《nε for any ε>0 and φ being a stabled smooth function compactly supported on R+,and under some hyothesis,they established the bound that Iwaniec,Luo and Sarnak[3]Sq(X)<<q 1/4X3/4+ε(1.2)For β=Sun[9]established an upper bound of S(X,α)=∑n～X λ(n)e(αnβ)unconditionally.In Sankaranarayanan[8],Sankaranarayanan improved the upper bound and obtain thatThe main purpose of this paper is to consider the case that[is variable and generalize the results in Sun[9]and Sankaranarayanan[8].The principal aim here is to prove Theorem 1.1 and Theorem 1.2.Theorem 1.1.Let λ(n)be the Liouville function.For any 0 ≠α∈R and 0<β ≤1,for any ε>0,we have where the implied constant depends only on ε.Remark 1.To prove Theorem 1.1,we will apply the method in Sankara-narayanan[8]and Sun[9].The main techniques we used is Vaughan’s identity and Perron’s formula.As β is variable,for working out the dependence of β,we must handle the terms with β in the denominator carefully,especially in the error terms.In addition,we must choose new parameters in some place to make the method work.When β=1/2,our results agree with Sankaranarayanan[8]and improved the result in Sun[9].Theorem 1.2.Let μ(n)be the Mobius function.For any 0 ≠α∈M and 0<β<1,for any ε>0,we have where the implied constant depends only on ε.Remark 2.When β=1/2,we get the upper bound estimate for our nonlinear exponential sums is |α|1/2X3/4.This agrees with(1.2)if we take α=-2(?)q for any positive integer q,and an=λ(n)or μ(n).This is the best results so far while conjecturally the exponent of X expects to be 1/2+ε.