Abstract
In this paper, we introduce one kind special Gauss sums; then, using the elementary and analytic methods to study the mean value properties of these kind sums, we obtain several exact calculating formulae for them.
1. Introduction
Let and be two positive integers with . For any integer and Dirichlet character , we write , where denotes the greatest integer less than or equal to . Then, we define the summations and aswhere and is any integer.
It is clear that is a generalization of the classical Gauss sums. In fact, if , then we have and
That is, becomes the classical Gauss sums . Gauss sums play a very important role in the study of analytic number theory, and many number theory problems are closely related to it, so some scholars have studied the properties of the classical Gauss sums and obtained many meaningful and interesting results; some of them can be found in [1–14]. For this reason, we think that it is necessary and meaningful to study the properties of .
On the contrary, is related to Dirichlet -function . In fact, if is an odd prime, and , then we have
Therefore, the study of the sums and has extensive theoretical significance and application values.
In this study, as an attempt in this direction, we first study the mean value properties of and . We shall use the elementary and analytic methods to prove the following several results.
Theorem 1. Let be an odd prime. Then, for any integer with , we have the identity
Theorem 2. Let be an odd prime. Then, we have the identitywhere denotes the solution of the congruence equation .
Theorem 3. Let be an odd prime. Then, for any nonprincipal character , we have the identitiesFrom Theorems 1 and 2, we can also deduce the following corollaries:
Corollary 1. Let be an odd prime. Then, we have the asymptotic formula
Corollary 2. Let be an odd prime. Then, we have the asymptotic formulaNotes. In fact, for any integer , using our methods and the results in [15], we can give an exact calculating formula for the general th power mean:However, when is large, the calculation is more complicated, so we do not consider it.
2. Several Lemmas
This section, we need to give some simple lemmas, which are necessary in the proofs of our theorems. Of course, the proofs of these lemmas also need some knowledge of elementary and analytic number theory, in particular, the properties of the character sums and the classical Gauss sums modulo . All these can be found in [16, 17], we do not repeat them. First, we have the following.
Lemma 1. Let be an odd prime and be a fixed integer. Then, for any nonprincipal Dirichlet character , we have the identitywhere denotes the classical Gauss sums.
Proof. First, we prove the first formula. Similarly, we can deduce the second one. For any positive integer , it is clear that there are two integers and such that . This number pair not only exists, but it is unique. In fact, we have and , where denotes the greatest integer . From these results and the properties of geometric series, we haveNow, we prove the third one. Taking in the second formula, note that and , from the properties of the classical Gauss sums, and we haveThis proves Lemma 1.
Lemma 2. Let be an odd prime. Then, we have the identity
Proof. First, for any integer with , from the properties of the quadratic Gauss sums, we havewhere and denotes Legendre’s symbol modulo .
From (14) and the properties of the complete residue system modulo , we haveApplying formula (15), we may immediately deduceThis proves the first formula in Lemma 2.
Similarly, note that the identityand we also haveThis proves Lemma 2.
Lemma 3. Let be an integer. Then, we have the identities
Proof. See [18] or Corollary 1 in [15].
3. Proofs of the Theorems
Applying three simple lemmas in Section 2, we can easily complete the proofs of our theorems. First, we prove Theorem 1. From the orthogonality of the characters modulo and Lemma 1 with , we have
This proves Theorem 1.
Proof of Theorem 2. From the orthogonality of the characters modulo and Lemma 1 with , we have
If , then and . From (21), we have
If , then , , and . From (21), we have
If , then , , and , . From (21), we have
If , then , , and . From (21), we have
Now, Theorem 2 follows from (22), (23), (24), and (25).
Now, we prove Theorem 3. For any nonprincipal character modulo , note that ; from the third formula in Lemma 1, we have
From (26) and Lemma 3, we have
Similarly, from (26) and Lemma 3, we also have
Now, Theorem 3 follows from (27)– (29).
This completes the proofs of all our results.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Authors’ Contributions
All authors have equally contributed to this work. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the N. S .F. (11771351) of P. R. China and WYUAS (2021RCKY0002).