(Department of Mathematics, Xian Jiaotong University, Xian 710049, China)

Abstract: In this paper, we deal with a Diophantine inequality involving a prime, two squares of primes and one k-th power of a prime which give an improvement of the result given by Alessandro Gambini.

Keywords: Diophantine inequality; Hardy-littlewood method; Davenport-heilbronn method

§1. Introduction

In 1946, Davenport and Heilbronn introduced an important variant of the Hardy-littlewood method. This method enabled them to establish the following theorem (for the proof of this theorem see [6] for details.)

Theorem 1.1.Suppose that s≥2k+1and that λ1,...λs are non-zero real numbers not all in rational ratio, and not all of the same sign when k is even. Then for every positive number η there exist integers x1,...,xs, not all zero, such that

In this paper, we deal with a Diophantine inequality involving a prime, two squares of primes and one k-th power of a prime which give an improvement of the result given by Alessandro Gambini. In his paper (see [1]) he proved the following result:

Theorem 1.2.Assume that1

has infinitely many solutions in prime variables p1, p2, p3, p4for any ε>0.

In this paper, we combine the method of [7] and [2] to improve the above result to the following theorem.

Theorem 1.3.Assume that1

has infinitely many solutions in prime variables p1, p2, p3, p4for any ε>0, wherewhen k ≥2, andwhen1

We point out that in the range 140/71

In this paper,In section 3,we will get the main term ofI(v,X;R)and a negligible error term.In section 4 and section 6, we prove all other terms are negligible. In section 7, we complete the proof of the Theorem 1.3. And note that the treatment ofI1(v,X,R) andI2(v,X,R) are similar, so we only treatI2(v,X,R).

§2. Notations and lemmas

Throughout this paper, the letterp, with or without subscript is a prime. AndLdenotes the logX.ηis any given real number. Letεandδbe arbitrarily small positive numbers. Letω(m) be the characteristic function of the set of primes, andρ2(m),ρ3(m),ρ4(m) constructed in [3]. We get the following result,ρ2(m)≥ω(m)≥ρ3(m)−ρ4(m).In addition to, we can get

Suppose thatXis a sufficiently large parameter. LetI2=[(δX)1/2,X1/2],Ik=[(δX)1/k,X1/k].Notebe any subinterval ofI2. Then it follows from the construction ofρ2(m),ρ3(m),ρ4(m)that

forj=2,3,4, whereκ2,κ3,κ4are positive constants satisfyingκ3−κ2κ4>0.Wrie

similar to the proof of Hua’s lemma, we can get the following result,

By the Euler summation formula, we can get

Moreover, by (2.2) we can get the following formula,

The following will be used later in the article:

(1) LetWhen 2

(2) LetWhen 1

(3)q1=X2/5be the denominator of a convergent toλ1/λ2.

(4)q2=X2/5be the denominator of a convergent toλ1/λ3.

We definewhere

By the second equation of (2.7), we obtain

We derive the following conclusion,

where N(X) denotes the number of solutions to the inequality

withSo we focus on the lower bound ofI(v,X;R). Now we divide the real line into four parts

whereThese sets are called the major arc, the intermediate arcs, the minor arc and the trivial arc, respectively. So we have

Now, we list some important lemmas, which play a very important role In the following article.

Lemma 2.1.( [7], equation (2.4)). We can get

Lemma 2.2.( [7], equation (2.5)). We can get

Lemma 2.3.( [4], Theorem 3.1). Let k ≥1be a real number. For0

where J(X,h)is the Selberg integral.

Lemma 2.4.( [4], Theorem 3.2). Let k ≥1be a real number and ε be an arbitrarily small positive constant. There exists a positive constant c1, such that

uniformly for X1−5/(6k)+ε ≤h≤H.

Lemma 2.5.( [2], Lemma 5). Let k>1, τ>0. We have

Lemma 2.6.( [2], Lemma 5). Let k>1, τ>0. We can get

Lemma 2.7.( [2], Lemma 4). Let k>1, τ>0. We can get the following formula

Lemma 2.8.( [2], Lemma 10).

Lemma 2.9.( [2], Lemma 7). For X−1≤|α|≤X−3/5we have

§3. The major arcs

In this section, we are going to mainly estimateJ1,J2,J3,J4andJ5, which are used in this paper. First we note that

Since the estimation ofJ1is similarly to the estimation ofJ1in [1], so we have

and the estimation ofJ4is similarly toJ3, so we omit it too. So the following, let us finish the estimation ofJ2,J3andJ5.

3.1. The estimation of J2

From the previous content of the paper, we can get

By (2.4), we have

So by H¨older‘s inequality we can get

Next we estimateB2. By (2.4) and (2.5), as well as by some estimates, we have

From the above estimate, we getJ2=o(η2X1+1/kL−2).So, we finish the estimation ofJ1.

3.2. The estimation of J3

By inequation (2.6), we can get

By PNT, inequation (2.4) and Lemma 1, we get the following useful conclusion

So, we get the following formulaJ3=o(η2X1/2+1/kL−2).So, we finish the estimation ofJ3.

3.3. The estimation of J5

Recall, we haveAnd by the arithmetic-geometric inequality, we get

Since the two summand are similar, we only treat one of them

By trivial estimate (2.2), (2.4) and the H¨older‘s inequality, we obtain

where we used Lemma 2.1, Lemma 2.3 and Lemma 2.4. Let’s estimateB5

where we used (2.2), Lemma 2.1 and Lemma 2.2. Thus we haveSo, we complete the estimation ofJ1,J2,J3,J4andJ5.

§4. The intermediate arcs

When 25/9

§5. The trivial arcs

By H¨older’s inequality and the trivial bound forSk(λ4α), Lemma 2.1 and Lemma 2.2, we can get

§6. The minor arcs

From Lemma 2.1, Lemma 2.2 and Lemma 2.8, letwe get

By the same way, we have

Note that for anywe have

whereZ1=2k1X1−3/20+ε,Z2=2k2X1/2−3/40+ε, andy=2rX−7/10+εfor some non-negative integersk1,k2,r. By, we obtain that forZ1≥X1−3/20+εand|S1(λ1α)|>Z1, there are coprime integersa1,q1satisfying

Similarly for forZ2≥X1/2−3/40+εand|S2(λ2α)|>Z2,there are coprime integersa2,q2satisfying

Letµ(·) denote Lebesgue measure. For simplicity, we take the notationAas a shortcut forS(Z1,Z2,y).

Lemma 6.1.Let A as a shortcut for S(Z1,Z2,y), We have

Proof.For α∈A and from previous discussion, there exist two pairs of coprime integers(a1,q1)and(a2,q2)such that

The above inequalities define an interval of α, say I=I(a1,q1,a2,q2). We have

Forwe have

By (6.1), (6.2) and (6.4), we have

where q is the denominator of a convergent toSo from the Legendre’s law of best approx-imation, we have |a2q1|≥q. By the same method, we have |a2(α)q1(α)−a2(α")q1(α")|≥q, for any pair α, α"having distinct associated products a2q1. So, there is at most one value of a2q1in the interval[rq,(r+1)q)for some positive integer r. By (6.5) and divisor argument, a2q1determines a1, a2, q1, q2to within Xε possibilities. From (6.4), we have

So we getNote that

So we have

So we finish the proof of the lemma.

By the above lemma and H¨older’s inequality, we have

Thus we get the following important inequalitywhich play a very important role in the following page.

§7. Proof of the theorem

By the above estimation ofJ1,J2,J3,J4andJ5, we can get the following inequation,

after calculation, we can getIt follow from (2.8) that

Recalling thatλ1/λ2,λ1/λ3are irrational,q1andq2are large enough denominator of convergent toλ1/λ2,λ1/λ3respectively, andfori=1,2. So whenq1,q2→∞, we haveX →∞,this impliesN(X)→∞. This complete the proof of the Theorem 1.3.

Acknowledgements

The authors would like to thank referees for providing helpful suggestions, which greatly improved the paper.