#### 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 [111]. 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 [38] 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).