Waring–Goldbach Problem of Even Powers in Short Intervals

Abstract

In this paper, we study the average behaviour of the representations of over short intervals for , where are prime numbers. This improves the previous results.

1. Introduction

Let and be integers; the classical Waring–Goldbach problem considers the representation of in the form

This problem has attracted the attention of many authors [1–11]. For sufficiently large integer , let be the exceptional set of expressing positive even integers up to in the form

Recently, Feng and Ma [2] established that , herewhere

Let

In our paper, we use the Hardy–Littlewood circle method and some techniques in the works of Languasco and Zaccagnini [3–8] to improve the results of Feng and Ma by studying the average behaviour of over short intervals , .

Theorem 1. Letbe a sufficiently large integer andbe integers. Then, there existsfor every, such thatas , uniformly for .

Theorem 1 implies that for a sufficiently large integer , every interval contains the integers which are sums of one prime square, two prime quartics, and one prime power in short intervals, where for . This improves Feng and Ma’s results obviously. For example, when , the exceptional set in Theorem 1 is smaller than in Feng and Ma’s results. Comparing the methods in Feng and Ma’s proof process, we replace finite sums with infinite series over primes and get a better result.

2. Preliminaries and Lemmas

Let be a sufficiently large integer, be an integer, , , , ,and

We have

We also set

From Montgomery ([11], p. 39), we can find that

Now, we need some lemmas as follows.

Lemma 1 (see [7], Lemma 3). Let,. Then, we haveuniformly for .

Lemma 2 (see [7], Lemma 1). .

Lemma 3 (see [7], Lemma 4). Letbe an arbitrarily small positive constant. Then, we haveuniformly for . Here, is a positive constant, which does not depend on .

Lemma 4. Letbe an arbitrarily small positive constant andbe a positive constant. Then,

Proof. This is Lemma 3.4 in [1] and Lemma 6 in [7].

Lemma 5. Letbe a arbitrarily small positive constant,be a fixed positive constant and. Let. Then,

Proof. This is Lemma 3.5 in [1] and Lemma 7 in [7].

Lemma 6. Letbe a fixed positive constant; we have

Proof. See the proof of Lemma 5 in [7] and Lemma 4.3 in [12] for .

3. Proof of Theorem 1

Let and for , where . From equation (5), we can get

We find it also convenient to set

Let ; we can get

Now, we begin to estimate these terms. First, we estimate . By the approximation , Lemma 1 , and equations (9) and (11), we can get

Next, we estimate .

From and , we can get

By Lemma 3 and equation (11), for every , then there exists such thatprovided that for . By Lemma 3, equation (11), and the Cauchy–Schwarz inequality, we obtain thatprovided that for . Hence, by equations (23) and (24), for every , then there exists such thatprovided that for .

By Lemma 4, equation (11), and , we haveand

Hence, by equations (27) and (28), we haveprovided that for . Thus, from equations (25) and (29), for every , there exists such thatprovided that for

Now, we estimate . By equations (11) and (19), Lemmas 3 and 6, and the Cauchy–Schwarz inequality, we obtain that for every , there exists such thatprovided that for .

Next, we estimate .

By Lemma 2 and , we obtain that

By Lemmas 4 and 5, equation (11), and the Cauchy–Schwarz inequality, we haveprovided that for .

Similarly, we also haveprovided that for .

Hence, by equations (34) and (35), we haveprovided that for .

By Lemma 2, equation (11), and , we have

By Lemmas 4 and 5 and the Cauchy–Schwarz inequality, we haveandprovided that for . Then, we haveprovided that for .

Hence, by equations (36) and (40), we haveprovided that for .

Now, we estimate . By Lemmas 4 and 5, equation (11), and a partial integration, we haveprovided that for .

Finally, we estimate . Clearly, by Lemma 5, equation (11), and Cauchy–Schwarz inequality, we haveprovided that for .

Completion of the proof. Let . By equations (19)–(37) and (40)–(43), we obtain that, for every , there exists , such thatprovided that with . From , for and every , there exists , such thatprovided that for . Using and equation (44), the last error term is . Thus, we haveuniformly for . Now, Theorem 1 follows.

Data Availability

The data used to support this study are available upon request from the corresponding author.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

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