LIU Rui, LIU Huafeng

(School of Mathematics and Statistics, Shandong Normal University, Ji′nan 250358, China)

Abstract：In this paper we consider the distribution of primes in kth-power progressions on average by the Hardy-Littlewood method. Let k ≥2 be an integer. We prove that there exists an effective positive constant δ >0 such that for all integers u ∈[1,xk] with at most O(xk-δ) exceptions, the lower bound of the mean value of Λ(nk +u) is GxL-k,where Λ is the von Mangoldt function and G is a certain non-effective constant with value depending on the Siegel zero. Our results improve previous results in the order of exceptions.

Keywords： Primes, kth-power progressions, the Hardy-Littlewood method

1 Introduction

It was established by Dirchlet that any linear polynomial represents infinitely many primes provided that the coefficients are co-prime. For any polynomial of higher degree,though it has long been conjectured, analogous statements are not known. The famous Bateman-Horn conjecture[1]predicts that iffis an irreducible polynomial in Z[x]satisfying

then

wherenpdenotes the number of solutions of the equationf(n)≡0(modp) in Z/pZ.When we specifyf(n) bynk+uwith an integerk≥2, we have

whereχis a Dirichlet character of orderk′=gcd(p-1,k).

In 2007, Baier and Zhao[2]studied (1) on average over a family of quadratic polynomials and proved that (1) holds true for all square-freeu≤ywith at mostO(yL-C) exceptions, wherex2L-A≤y≤x2for anyA> 0. More precisely, their result states the following.

Theorem 1.1GivenA,B,C>0, we have, forx2L-A≤y≤x2, that

holds for all square-freeunot exceedingywith at mostO(yL-C) exceptions, where

Later Baier and Zhao[3]extended the range ofyand removed the restriction of square-free numbers onu. But the order of the exceptions forunot exceedingyis still at mostO(yL-C).

In 2013, Foo and Zhao[4]studied the asymptotic formula in (1) on average over a family of cubic polynomials and established similar results. Their result is as follows.

Theorem 1.2GivenA,B,C>0, we have, forx3L-A≤y≤x3, that

holds for all square-freeunot exceedingywith at mostO(yL-C) exceptions, where

withnpbeing the number of solutions to the equationn3+u≡0(modp) in Z/pZ.

Now we focus on the order of exceptions. If we do not require the asymptotic formula(1),the order of exceptions can be improved. In 2011,Lü and Sun[5]improved the order of exceptions for the quadratic polynomialn2+uand showed the following Theorem 1.3.

Theorem 1.3There exist effective positive constantsδ′,δ′′such that, forx2-δ′≤y≤x2,

holds true for all integers 1 ≤u≤ywith at mostO(y1-δ′′) exceptions, whereGis defined by (6) withk=2.

In this paper, we shall generalize Lü and Sun′s result to the polynomialnk+uwithk≥2. Our results are stated as follows.

Theorem 1.4Letk≥2 be an integer andGbe defined by (6).

follows for all integers 1 ≤u≤xkwith at mostO(xk-δ) exceptions, whereδis an effective computable positive constant.

From Theorem 1.4, we can get the following corollary.

Corollary 1.1Letk≥2 be an integer andGbe defined by (6). Forxk-δ1≤y≤xk,

follows for all integers 1 ≤u≤ywith at mostO(y1-δ2) exceptions, whereδ1,δ2are effective computable positive constants.

It is easy to check that whenk=2,3, the orders of the exceptions now are better than ones in Baier and Zhao′s result, and in Foo and Zhao′s result, respectively. In 2018, Zhou[6]considered the asymptotic formula (1) over the polynomialnk+uon average, and improved the results on range ofyin [3-4]. But Zhou′s result also implies that the order of exceptions isO(yL-C)with a suitableC>0. Our results also improve Zhou′s result on the order of exceptions.

As usual, we abbreviate e2πiαtoe(α) and logxtoL. The letterpdenotes a prime number.ρ=β+iγdenotes the generic zero of anL-function. Denote byφ(n)andμ(n)the Euler′s function and the M?bius′function, respectively. The constantsc1,c2,···,which may depend onk, are effectively computable positive constants.

2 Proof of Theorem 1.4

We begin with the point that

Then for any fixed 1 ≤u≤xkwe can rewriteR(x,u) as

where

Let

Therefore we have

Then we find that this problem is similar to representing an integeruas the sum of a prime and ak-th power, i.e.,

Whenk=2,many scholars has studied this problem and we refer to[7,8,9]for details.In 1992, Zaccagnini[10]first extended this problem to the general casek≥2. The ideas of Zaccagnini extend the works of Vaughan[11]and Montgomery and Vaughan[12], which causes the main difficulty to handle the singular series. The most novel feature of Zaccagnini′s work is the application of estimates for character sums over polynomial values to conquer these difficulties. So based on this result we may study it briefly in this paper.

Let

and

Then we denote byM(a,q) the major arcs

whereq≤Pand (a,q)=1. The major arcsMand the minor arcsC(M) now can be defined as

Then we can writeR(x,u) as follows:

To handlingR1(x,u) andR2(x,u), we need the following lemma, which shows an approximation and an estimate forFk(α) on the major arcs and on the minor arcs,respectively.

Lemma 2.1Let (a,q)=1. Then we have

where

IfP

for a suitable constantθ=θ(k)>0.

ProofWe can find this lemma in Lemma 5.1 in [10]. The former inequality can be proved as Lemma 3 in [13]. The latter one follows from Weyl′s inequality, e.g.(see Lemma 2.4 in [14]).

Now we begin to estimate the contribution from the minor arcs,which is standard.Applying Bessel′s inequality and the prime number theorem we have

Then by the definition ofC(M) and Lemma 2.1, we have

Therefore from (10) we can deduce that

except for at mostO(xkP-θL) exceptions.

Next we turn to estimate the integral on the major arcs. By orthogonality we have

where

We defineP-excluded zeros as those zeros of the functionsL(s,χ), whereχis any primitive character moduloq,q≤P, lying in the region

excluding the possible Siegel zero. Then theP-excluded characters can be defined as the primitive charactersχ(modr),r≤P, for whichL(ρ,χ)=0,ρbeing aP-excluded zero. TheP-excluded moduli are the moduli of theP-excluded characters.

To approximateS(α) we defineW(χ,λ) in the following way.

(1) Ifχ=χ0,q, then

where

(2) Ifχ(modq) is induced byχ?∈E ∪J, thenχ=χ0,qχ?and

where

and

(3) In all other cases,

Then we can obtain

where

and

From Lemma 2.1, we have

where

Inserting (12) and (13) intoR1(x,u) we can get

Now the remaining work is to estimate these termsS1,··· ,S4.

The methods of handlingS1andS4are standard. By a standard argument we can get

where

By Cauchy′s inequality and the prime number theorem, we have

By the argument in Reference [10], we have

where

Following the arguments of the proof of Theorem 7 in [15] and applying Lemma 4.8 in[16] we can get

From Section 10 in [10], we get

where

Putting (14), (15), (19), (20) and (21) into together we have

Following the arguments in Section 14 in [10], with at mostO(xk-δ3)exceptions for anyuwe have

where we have separated the possibly existent Siegel zero, and the second term including the Siegel zero is to be deleted wheneverdoes not exist.

We also have that for allu≠mkwith at mostO(xk-δ3)exceptions,

wherer∈{r≤,ris an exluded or Siegel modulus},χis a primitive character modulorand

Moreover, we have

With the help of these estimates we can get

Recalling the definition ofηin (16), we can chooselsuch that

Note that

Thus we get that

with at mostO(xk-δ3)exceptions for anyu.

Then Theorem 1.4 follows from (11) and (28).