Evaluations of a Weighted Average of Gauss Sums

Shao, Wen-Kai; He, Yuan

doi:https://doi.org/10.1155/2021/6206329

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Classical Problems in Algebra and Number Theory

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Research Article | Open Access

Volume 2021 | Article ID 6206329 | https://doi.org/10.1155/2021/6206329

Evaluations of a Weighted Average of Gauss Sums

Wen-Kai Shao¹and Yuan He²

Academic Editor: Yuan Yi

Received23 May 2021

Accepted17 Jun 2021

Published30 Jun 2021

Abstract

In this paper, we perform a further investigation for a weighted average of Gauss sums. By making use of some properties of the cotangent function and the Bernoulli polynomials, we explicitly evaluate the weighted average of Gauss sums in terms of the special values of Dirichlet -functions at positive integers.

1. Introduction

Let be a positive integer, and let be a Dirichlet character modulo . The Gauss sum associated with is defined for an arbitrary integer (see, e.g., Section 8.5 in [1]). where is the imaginary unit. In particular, the case in (1) is usually denoted by

It is well known that various properties and applications of appear in many books and papers. For example, an important result of is as follows (see, e.g., Theorem 8.15 in [1]):if is a primitive Dirichlet character modulo . We here mention [2–5] for further explorations of (3). Moreover, it is also demonstrated that some weighted averages of have good distribution properties when is a nonprimitive Dirichlet character modulo . For example, let be a positive integer and let be a positive real number, and Yi and Zhang [6] used the estimate for character sums and the method of trigonometric sums to study the mean value of with the weight of the inversion of , where denotes the Dirichlet -functions defined for a complex number and a Dirichlet character modulo by the series as follows:and gave a sharper asymptotic formula. After that Liu and Zhang [7] used some properties of primitive Dirichlet character and some mean value formulas of Dirichlet -functions to study the high-order mean value of products of and the generalized Bernoulli numbers and obtained an interesting asymptotic formula. We also refer to [8] for a further investigation for the high-order mean value of products of and the generalized Bernoulli numbers. More recently, Alkan [9] used a special evaluation method for Dirichlet -functions to consider the following weighted average of Gauss sums:where is a real-valued function defined on the interval and depicted that the weighted averages of Gauss sums and and the character values at positive integers can be well approximated by linear combinations of the algebraic parts of special values of Dirichlet -functions under correct parity conditions (see, e.g., Theorems 1, 2, and 3 in [9]). The special evaluation method for Dirichlet -functions posed by Alkan (see Theorem 4 in [9]) states that if for a positive integer , then for a nonprincipal even Dirichlet character modulo ,and for an odd Dirichlet character modulo ,where the coefficients for a positive integer with and the coefficients for a nonnegative integer with can be computed recursively by using the Bernoulli numbers.

Motivated and inspired by the work of Alkan [9], in this paper, we use some properties of the cotangent function and the Bernoulli polynomials to study the above weighted average of Gauss sums and determine the coefficients appearing in (6) and (7) which can be explicitly evaluated by the binomial coefficients. The precise statements of our results are as follows.

Theorem 1. Let for a positive integer . Then, for a nonprincipal even Dirichlet character modulo ,and for an odd Dirichlet character modulo ,where denotes the greatest integer less than or equal to real number , and the sum on the right hand side of (8) vanishes when .

It becomes obvious that Theorem 1 gives the following explicit evaluations of the coefficients appearing in Alkan’s [9] formulas (6) and (7).

Corollary 1. Let be positive integers, and let be a nonnegative integer. Suppose that are the same as those in (6) and are the same as those in (7). Then,

This paper is organized as follows. In the second section, we present some auxiliary results. The third section concentrates on the features that have contributions to the proof of Theorem 1.

2. Some Auxiliary Results

Before giving the proof of Theorem 1, we need the following auxiliary results.

Lemma 1. Let be positive integers with , and let be a nonprincipal Dirichlet character modulo . Then,where is given for positive integers with bywith being the Stirling numbers of the first kind.

Proof. (see Theorem 2.2 in [10] for details).

Remark 1. It is worth noticing that the finite trigonometric sum on the left hand side of (11) was also studied by Zhang and Lin [11], where a nice connection between the Dirichlet -functions at even positive integers and the finite trigonometric sum on the left hand side of (11) is established, and some interesting identities involving finite trigonometric sums are deduced, and a new proof for the mean square value formula of Dirichlet L-functions showed in [12, 13] is also presented.

Lemma 2. Let be a positive integer, and let be a real function defined on a positive integer . Then,where is the Kronecker delta given by or 0 according to or , respectively, and is the periodic zeta function given for a real number and a complex number by

Proof. (see Equation (2.28) in [10] for details).

Lemma 3. Let be a positive integer. Then, if is an even Dirichlet character modulo , then for a positive integer ,and if is an odd Dirichlet character modulo , then for a nonnegative integer ,where are the Bernoulli polynomials defined by the generating function (see, e.g., [1, 14])

Proof. It is obvious from (4) to see thatwhere is the Hurwitz zeta function given with a real number and a complex number byBy taking in (18), we get that for a positive integer ,Since the Hurwitz zeta function at positive integers satisfies the reflection formula (see, e.g., Section 4 in [15] or equation (25) in [16]),From (20) and (21), we obtain that for a positive integer , if , thenApplying Lemma 2 to the right side of (22), we claim that for a positive integer , if , thenWe know from Theorem 12.13 in [1] that for a nonnegative integer ,which impliesHence, by inserting (25) into (23), in view of (1), we get that for a positive integer , if , thenIt follows from (26) that (15) and (16) hold true for a positive integer . We next prove that the case in (16) is complete. In fact, since (see, e.g., p. 266 in [1]), by the familiar geometric sum stated in Theorem 8.1 in [1] and the property of character sums described in Theorem 6.10 in [1], we discover that for an odd Dirichlet character modulo ,On the other hand, by taking in Lemma 1, in light of for a nonnegative integer (see, e.g., p. 214 in [17]), we obtain that for an odd Dirichlet character modulo ,Thus, combing (27) and (28) gives that for an odd Dirichlet character modulo ,which means (16) holds true for the case . This completes the proof of Lemma 3.

Remark 2. Since the Bernoulli polynomials can be expressed by the Bernoulli numbers in the following way (see, e.g., Theorem 12.12 in [1]),where is the -th Bernoulli number; by applying (30) to Lemma 3, one can easily get that if is a positive integer and such that r and n have the same parity, and if ,where is given for a nonnegative integer byFormula (31) was firstly discovered by Alkan Theorem 1 in [18] using the Fourier expansions of the Bernoulli periodic functions and some properties of character sums and is also a key ingredient for the proofs of formulas (6) and (7). We here refer to Theorem 2 in [19] for an extension of (31). For a different proof of Lemma 3, one can consult to Theorem 1.3 in [20], where the author used the function equation for the Hurwitz zeta function to give the proof.

3. The Proof of Theorem 1

We are now in a position to provide the detailed proof of Theorem 1. It is well known that the Bernoulli polynomials satisfy the following difference equation (see, e.g., Theorem 12.14 in [1]):and the following addition formula (see, e.g., p. 275 in [1]):

Hence, we obtain from (33) and (34) that for a positive integer ,

It follows from (1), (35), and (see, e.g., p. 266 [1]) that

Since for a nonprincipal Dirichlet character modulo , if is a nonprincipal Dirichlet character modulo , we havewhere we used the symmetric relation of the Bernoulli polynomials (see, e.g., p. 274 [1]) as follows:

Moreover, the familiar geometric sum implies that

It follows from (1), (36), (37), and (39) that if is a nonprincipal even Dirichlet character modulo , thenand if is an odd Dirichlet character modulo , then

Applying Lemma 3 to the right hand sides of (40) and (41), we get that if is a nonprincipal even Dirichlet character modulo , thenand if is an odd Dirichlet character modulo , then

Now, (8) and (9) follow from (42) and (43), respectively. This concludes the proof of Theorem 1.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2021 Wen-Kai Shao and Yuan He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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