Sums of One Prime Power and Four Prime Cubes in Short Intervals

Abstract

Let be an integer. In this study, we derive an asymptotic formula for the average number of representations of integers in short intervals, where are prime numbers.

1. Introduction

Let be integers with . The Waring–Goldbach problem for unlike powers of primes concerns the representation of as the formis classical. These topics have attracted mathematicians’ attention.

In 1953, Prachar [1] considered the representation of as the formand he obtained the exceptional set is . This result has been improved by Bauer [2], Bauer [3], and Zhao [4], and the latest result is . For general , the best result was given by Hoffman and Yu [5] which is where .

In 1961, Schwarz et al. [6] also considered the representation of as the formand they obtained the exceptional set is for any fixed . Later, Brüdern [7] improved it to .

Recently, Feng and Ma [8] considered a special case,with . Let be the number of positive even integers up to which cannot be written in the form (4). They established that , herewhere

Let

In this study, we want to reconsider the result of Feng and Ma by studying the average behaviour of over short intervals and .

Theorem 1. Let be integers. Then, for every , there exists , such thatas , uniformly for and is Euler’s function.

Comparing the result in Feng and Ma, from Theorem 1 we can say that, for sufficiently large, every interval of size large than contains the expected amount of integers which are a sum of one prime power and four prime cubes.

Assuming that the Riemann Hypothesis (RH) holds, we can further improve the size of .

Theorem 2. Let be integers and assume the Riemann Hypothesis holds. Then, there exists a suitable positive constant such thatas , uniformly for . Here, means and is Euler’s function.

The proofs of Theorems 1 and 2 use the original Hardy–Littlewood circle method and the strategies adopted in the works of Languasco and Zaccagnini [9–11].

2. Preliminaries

In this study, we assume that is a sufficiently large integer. Let and be an integer,

We have

We also set

From Montgomery [12], p. 39, we have

Now, we need some lemmas as follows.

Lemma 3 ([11], Lemma 3]). Let be an integer. Then, we have

Lemma 4 ([10], Lemma 2]). Let be a sufficiently large integer and be an integer. Then,where runs over the nontrivial zeros of the Riemann-zeta function .

Lemma 5 ([7], Lemma 4]). Let. Then,uniformly for .

Lemma 6 ([11], Lemma 1]). Let be an integer and be an arbitrarily small positive constant. Then, there exists a positive constant , which does not depend on , such thatuniformly for . Assuming RH holds we obtainuniformly for .

Lemma 7 ([9], Lemma 5]). Let be integers with . There exists a suitable positive constant , such that

Lemma 8 ([9], Lemma 6]). Let and . Then, we have

Lemma 9 ([9], Lemma 7]). Let and . We also let . Then, we have

3. Proof of Theorem 1

Let withwhere will be chosen later. Recalling (7), we can write

We find it also convenient to set

Let , we can obtain

Now, we need to estimate these terms.

3.1. Estimate of

From the approximation , Lemma 5, and (11) we can obtain

3.2. Estimate of

From the identity , (24), and , we obtain

Using (11) and (27), we obtain

By Lemma 6 we can obtain, for every , there exists such thatprovided that , i.e., . By (11) and (29) and the Cauchy–Schwarz inequality we have, for every , there exists , such thatprovided that . By Lemma 7, (11), (29), and the Cauchy–Schwarz inequality we also have, for every , there exists such thatprovided that . Hence, by (28)–(31), we finally obtain that for every , there exists such thatprovided that .

3.3. Estimate of

Now, we estimate . By (13), Lemmas 6 and 7, and the Cauchy–Schwarz inequality, we have, for every , there exists such thatprovided that .

3.4. Estimate of

From Lemma 3 and , we have

Then, we have

By (13), Lemmas 8, and 9, we haveprovided that . Similarly, we have

By (35)–(37), we haveprovided that .

3.5. Estimate of

By Lemmas 3, 8, 9, (13), and a partial integration, we haveprovided that .

3.6. Estimate of

Clearly, by Lemma 9 and (13), we haveprovided that .

3.7. Completion of the Proof

Let . By (26)–(40) we can obtain, for every , there exists , such thatprovided that . We choose in (22). Then, for every , we have for every , there exists , such thatprovided that . We note that for . Then, we have for every , there exists , such thatprovided that and for . Using and (41), the last error term is . Thus,uniformly for . Now, Theorem 1 follows.

4. Proof of Theorem 2

Let , , and be an integer. We recall that we set for brevity. From now on we assume that RH holds, we may write

In this conditional case, we can simplify the proof. Recalling Lemma 4 and (24), we have

Now, we can evaluate these terms.

4.1. Evaluation of

Using Lemma 5, a direct calculation gives

4.2. Evaluation of

By (27), we have

Let

Using Lemma 6, (13), and integration by parts, we have

Then, we have

By the Cauchy–Schwarz inequality, (11), (13), and (50), we have

By the Cauchy–Schwarz inequality, (11), (13), and (50), we have

Summing up by (48)–(54), we havefor every .

4.3. Evaluation of

Using the Cauchy–Schwarz inequality, (13), and Lemma 6, we can obtain

4.4. Evaluation of

Clearly, of Section 3.4. So, we can obtainprovided that .

4.5. Evaluation of

Clearly, of Section 3.5. So, we can obtainprovided that .

4.6. Completion of the Proof

By (46) and (47) and (54)–(57), there exist such that for ,which is an asymptotic formula for . From for and , we can obtain the following:

Using and (58), the last error term is dominated by all of the previous ones. Thus, Theorem 2 follows.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Gen Li wrote Section 2 and Section 4, Xianjiu Huang wrote Section 1 and some English expressions in Sections 2–4, Xiaoming Pan wrote Sections 3.1–3.3, and Li Yang wrote Sections 3.4–3.7, abstract, and references.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant no. 11761048), Natural Science Foundation of Jiangxi Province for Distinguished Young Scholars (Grant no. 20212ACB211007), and Natural Science Foundation of Jiangxi Province (Grant no. 20224BAB201001).