On the Error Term for the Number of Solutions of Certain Congruences

(School of Mathematics,Hefei University of Technology,Hefei 230009,China)

§1.Introduction

Letf(x)Suppose thatLis the splitting field offover Q with Galois groupG=Gal(L/Q).IfGis Abelian,we call the fieldLis Abelian and thatf(x)is an Abelian polynomial.Otherwise we callf(x)a non-Abelian polynomial.Letr(n)denote the number of solutionsxof the congruencef(x)≡0(modn)satisfying 0≤x＜n.We introduce theL-function associated tor(n),

It is a classical problem to study the functionr(n).In 1952 Erd?s[1]proved that

and the lower bound

It is hard to refine Erd?s’result.Until 2001,Fomenko[2]proved the asymptotic formulae

whereC(f)is a positive constant depending onf.

In[2],Fomenko also proved that for any Abelian polynomialf(x)of degreem,

for a certain positive constantBand any fixedβ＜.?By introducing the￠L(zhǎng)anglands functionality to this problem,Kim[6]improved the error termOxexp(?B(logx)β)intoO(x1?3/(m+3)+ε)for any fixedε＞0.

In addition,in[6]it was shown that for any non-Abelian polynomialf(x)of degreem≥2,

whereαis the residue ofL(s)which is defined in(1.1)at its simple poles=1.Recently based on Kim’s method,LYU[7]improved Kim’s result and obtained the sharper error termO?x1?3/(m+6)+ε￠.

Motivated by[2-4],the aim of the present paper is to investigate?(x)in mean square and we shall prove the following theorem

Theorem 1Let?(x)be defined in(1.3).For any non-Abelian polynomialf(x)of degreem≥2 and suppose thatm≥3.Then we have

In fact the proof shows that

form=2.

RemarkForm=2,3,the result of Theorem 1 is as good as that obtained under the Riemann Hypothesis on DirichletL-functions on average.

NotationsThe Vinogradov symbolA?Bmeans thatBis positive and the ratioA/Bis bounded.The letterεan arbitrary small positive number,not the same at each occurrence.

§2.Proof of Theorem 1

To prove Theorem 1,we need the following lemmas.

Lemma 2.1LetKbe a Galois extension of degreenover Q andL(s)be defined in(1.1).Then we have

whereU(s)denotes a Dirichlet series,which is absolutely and uniformly convergent for Re(s)＞andζ(s,K)denotes the Dedekind zeta function of the fieldK

ProofThis follows immediately by the arguments(see pages 319-320)in[7].

Lemma 2.2LetKbe an algebraic number field of degreem.Then for any fixedε＞0

ProofBy Lemma 2.5 in[8]and the Phragmén-Lindel?f principle for a strip(see e.g.Theorem 5.53 in Iwaniec and Kowalski[5],this lemma follows.

Now we begin to complete the proof of Theorem 1.By the definition ofr(n),we have

From Lemma 3 in[1],it is known that the functionr(n)is multiplicative and satisfies

for natural numberl＞0,whereDis the discriminant off(x).Then we have

whereω(n)denotes the number of distinct prime divisors ofnandτ(n)is the divisor function.

Let

From(2.4),(1.1)and Perron’s formula(see Proposition 5.54 in[5]),we get

By the propertyL(s)only has a simple pole ats=1 for＞and Cauchy’s residue theorem,we have

Then in order to prove(1.4)in Theorem 1,we shall prove the following results.

To go further,we get

By(2.10)～(2.11)and(2.1),

where we have used lemmas 2.1 and 2.2.

By Lemma 2.2,we have

which yields

The inequalities(2.7)～(2.8)immediately follow from(2.9),(2.12)～(2.13).That is,we have

Form=2,by takingT=,the detail of the proof of(1.5)is similar to that of(1.4),hence we omit it here.Then the proof of Theorem 1 is completed.

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