# Evaluations of a Weighted Average of Gauss Sums

Authors: Wen-Kai Shao; Yuan He
Journal: Journal of Mathematics
 (2021)

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## 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 DirichletL-functions at positive integers.


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## Body

## 1. Introduction

Letq be a positive integer, and let χ be a Dirichlet character modulo q. The Gauss sum associated with χ is defined for an arbitrary integer n (see, e.g., Section 8.5 in [1]). (1)Gn,χ=∑r=1qχre2πirn/q,where i is the imaginary unit. In particular, the case n=1 in (1) is usually denoted by(2)τχ=∑r=1qχre2πir/q.It is well known that various properties and applications ofτχ appear in many books and papers. For example, an important result of τχ is as follows (see, e.g., Theorem 8.15 in [1]):(3)τχ=q,if χ is a primitive Dirichlet character modulo q. We here mention [2–5] for further explorations of (3). Moreover, it is also demonstrated that some weighted averages of τχ have good distribution properties when χ is a nonprimitive Dirichlet character modulo q. For example, let k be a positive integer and let m 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 τχm with the weight of the inversion of L1,χ2k, where Ls,χ denotes the Dirichlet L-functions defined for a complex number s and a Dirichlet character χ modulo q by the series as follows:(4)Ls,χ=∑n=1∞χnns,ℜs>1,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 L-functions to study the high-order mean value of products of τχ 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 τχ and the generalized Bernoulli numbers. More recently, Alkan [9] used a special evaluation method for Dirichlet L-functions to consider the following weighted average of Gauss sums:(5)Aχω=1q∑r=1qωrqGr,χ,where ω⋅ is a real-valued function defined on the interval 0,1 and depicted that the weighted averages of Gauss sums Aχω and iAχω and the character values χm at positive integers m can be well approximated by linear combinations of the algebraic parts of special values of Dirichlet L-functions under correct parity conditions (see, e.g., Theorems 1, 2, and 3 in [9]). The special evaluation method for Dirichlet L-functions posed by Alkan (see Theorem 4 in [9]) states that if ωx=xm for a positive integer m, then for a nonprincipal even Dirichlet character χ modulo q,(6)Aχω=C2,mL2,χπ2+C4,mL4,χπ4+⋯+Cm,mLm,χπm,2|m,C2,m×L2,χπ2+C4,m×L4,χπ4+⋯+Cm−1,m×Lm−1,χπm−1,2∤m,and for an odd Dirichlet character χ modulo q,(7)Aχω=D1,mL1,χπ+D3,mL3,χπ3+⋯+Dm,mLm,χπm,2∤m,D1,m×L1,χπ+D3,m×L3,χπ3+⋯+Dm−1,m×Lm−1,χπm−1,2|m,where the coefficients C2k,m,C2k,m×∈ℚ for a positive integer k with 2k≤m and the coefficients D2j+1,m,D2j+1,m×∈iℚ for a nonnegative integer j with 2j+1≤m 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.
Letωx=xm for a positive integer m. Then, for a nonprincipal even Dirichlet character χ modulo q,(8)Aχω=−1m+1∑j=1m/2m+12j−1j2j!⋅L2j,χπ2j⋅22j−1,and for an odd Dirichlet character χ modulo q,(9)Aχω=−im+1∑j=0m−1/2m+12j+1−1j2j+1!⋅L2j+1,χπ2j+1⋅22j,where x denotes the greatest integer less than or equal to real number x, and the sum on the right hand side of (8) vanishes when m=1.It becomes obvious that Theorem1 gives the following explicit evaluations of the coefficients appearing in Alkan’s [9] formulas (6) and (7).Corollary 1.
Letkandm be positive integers, and let j be a nonnegative integer. Suppose that C2k,mandC2k,m× are the same as those in (6) and D2j+1,m,D2j+1,m× are the same as those in (7). Then,(10)C2k,m=C2k,m×=−−1km+1m+12k2k!22k−1,D2j+1,m=D2j+1,m×=−i−1jm+1m+12j+12j+1!22j.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 Theorem1.


## 2. Some Auxiliary Results

Before giving the proof of Theorem1, we need the following auxiliary results.Lemma 1.
Letqandn be positive integers with q≥2, and let χ be a nonprincipal Dirichlet character modulo q. Then,(11)∑r=1q−1χrcotnπrq=∑k=1nqkAn,kπkLk,χ+χ−1∑k=1n−1kqkAn,kπkLk,χ,where An,k is given for positive integers n,k with 1≤k≤n by(12)An,k=in−kk−1!2k∑l=knnl2lsl,kl−1!,with sl,k 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 L-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.
Letn be a positive integer, and let θr be a real function defined on a positive integer r. Then,(13)∂n−1∂an−1cotπa|a=θr/q=δ1,ni+2nπn−1in−2Fθrq,1−n,where δ1,n is the Kronecker delta given by δ1,n=1 or 0 according to n=1 or n≠1, respectively, and Fa,s is the periodic zeta function given for a real number a and a complex number s by(14)Fa,s=∑n=1∞e2πinans,ℜs>1.Proof.
(see Equation (2.28) in [10] for details).Lemma 3.
Letq be a positive integer. Then, if χ is an even Dirichlet character modulo q, then for a positive integer n,(15)L2n,χ=−1n−1π2n⋅22n−1q⋅2n!∑r=1qB2nrqGr,χ,and if χ is an odd Dirichlet character modulo q, then for a nonnegative integer n,(16)L2n+1,χ=−1niπ2n+1⋅22nq⋅2n+1!∑r=1qB2n+1rqGr,χ,where Bnx are the Bernoulli polynomials defined by the generating function (see, e.g., [1, 14])(17)textet−1=∑n=0∞Bnxtnn!,t<2π.Proof.
It is obvious from (4) to see that(18)Ls,χ=∑m=0∞∑l=1qχqm+lqm+ls=1qs∑l=1qχlζs,lq,where ζs,x is the Hurwitz zeta function given with a real number x>0 and a complex number s by(19)ζs,x=∑n=0∞1n+xs,ℜs>1.
By takings=n in (18), we get that for a positive integer n,(20)Ln,χ=1qn∑l=1qχlζn,lq=χ−1qn∑l=1qχlζn,1−lq.
Since the Hurwitz zeta function at positive integers satisfies the reflection formula (see, e.g., Section 4 in [15] or equation (25) in [16]),(21)ζn,a+−1nζn,1−a=−1n−1πn−1!⋅∂n−1∂an−1cotπa,n≥2.
From (20) and (21), we obtain that for a positive integer n≥2, if χ−1=−1n, then(22)Ln,χ=−1n−1π2qn⋅n−1!∑l=1qχl∂n−1∂an−1cotπa|a=1/q.
Applying Lemma2 to the right side of (22), we claim that for a positive integer n≥2, if χ−1=−1n, then(23)Ln,χ=in−2−1n−1πn⋅2n−1qn⋅n−1!∑l=1qχlFlq,1−n.
We know from Theorem 12.13 in [1] that for a nonnegative integer n,(24)ζ−n,x=−Bn+1xn+1,which implies(25)Flq,1−n=∑j=0∞∑r=1qe2πiqj+rl/qqj+r1−n=−1nq1−n∑r=1qe2πirl/qBnrq.
Hence, by inserting (25) into (23), in view of (1), we get that for a positive integer n≥2, if χ−1=−1n, then(26)Ln,χ=in−2−1nπn⋅2n−1q⋅n!∑r=1qBnrqGr,χ.
It follows from (26) that (15) and (16) hold true for a positive integer n. We next prove that the case n=0 in (16) is complete. In fact, since B1x=x−1/2 (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 χ modulo q,(27)∑r=1qB1rqGr,χ=∑l=1q−1χl∑r=1qrq−12e2πirl/q=1q∑l=1q−1χl∑r=1q−1re2πirl/q=∑l=1q−1χl1e2πil/q−1=12∑l=1q−1χleπil/q+e−πil/qeπil/q−e−πil/q−1=12i∑l=1q−1χlcotπlq.
On the other hand, by takingn=1 in Lemma 1, in light of sn,n=1 for a nonnegative integer n (see, e.g., p. 214 in [17]), we obtain that for an odd Dirichlet character χ modulo q,(28)∑r=1q−1χrcotπrq=2qπL1,χ.
Thus, combing (27) and (28) gives that for an odd Dirichlet character χ modulo q,(29)∑r=1qB1rqGr,χ=qπiL1,χ,which means (16) holds true for the case n=0. 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]),(30)Bnx=∑k=0nnkBkxn−k,n≥0,where Bk is the k-th Bernoulli number; by applying (30) to Lemma 3, one can easily get that if n is a positive integer and r∈0,1 such that r and n have the same parity, and if χ−1=−1r,(31)Ln,χ=−1r+1inπn⋅2n−1q⋅n!∑k=02n/2nkBkSn−k,χ,where Sm,χ is given for a nonnegative integer m by(32)Sm,χ=∑r=1qrqmGr,χ.
Formula (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 Theorem1. It is well known that the Bernoulli polynomials satisfy the following difference equation (see, e.g., Theorem 12.14 in [1]):(33)Bnx+1−Bnx=nxn−1,n≥1,and the following addition formula (see, e.g., p. 275 in [1]):(34)Bnx+y=∑k=0nnkBkxyn−k,n≥0.Hence, we obtain from (33) and (34) that for a positive integer m,(35)xm=Bm+1x+1−Bm+1xm+1=1m+1∑j=0m+1m+1jBjx−Bm+1x=1m+1∑j=0mm+1jBjx.It follows from (1), (35), and B0x=1 (see, e.g., p. 266 [1]) that(36)∑r=1qrqmGr,χ=1m+1∑j=0mm+1j∑r=1qBjrq∑l=1qχle2πilr/q=1m+1∑j=1mm+1j∑r=1qBjrq∑l=1qχle2πilr/q+1m+1∑l=1qχl∑r=1qe2πilr/q.SinceG±q,χ=0 for a nonprincipal Dirichlet character χ modulo q, if χ is a nonprincipal Dirichlet character modulo q, we have(37)∑r=1qBjrq∑l=1qχle2πilr/q=∑r=1q−1Bjrq∑l=1qχle2πilr/q=∑r=1q−1Bj1−rq∑l=1qχle2πilq−r/q=−1jχ−1∑r=1qBjrq∑l=1qχle2πilr/q,where we used the symmetric relation of the Bernoulli polynomials (see, e.g., p. 274 [1]) as follows:(38)Bn1−x=−1nBnx,n≥0.Moreover, the familiar geometric sum implies that(39)∑l=1qχl∑r=1qe2πilr/q=0.It follows from (1), (36), (37), and (39) that if χ is a nonprincipal even Dirichlet character modulo q, then(40)∑r=1qrqmGr,χ=1m+1∑j=12|jmm+1j∑r=1qBjrqGr,χ=1m+1∑j=1m/2m+12j∑r=1qB2jrqGr,χ,and if χ is an odd Dirichlet character modulo q, then(41)∑r=1qrqmGr,χ=1m+1∑j=12∤jmm+1j∑r=1qBjrqGr,χ=1m+1∑j=0m−1/2m+12j+1∑r=1qB2j+1rqGr,χ.Applying Lemma3 to the right hand sides of (40) and (41), we get that if χ is a nonprincipal even Dirichlet character modulo q, then(42)∑r=1qrqmGr,χ=−qm+1∑j=1m/2m+12j−1j2j!⋅L2j,χπ2j⋅22j−1,and if χ is an odd Dirichlet character modulo q, then(43)∑r=1qrqmGr,χ=−qim+1∑j=0m−1/2m+12j+1−1j2j+1!⋅L2j+1,χπ2j+1⋅22j.Now, (8) and (9) follow from (42) and (43), respectively. This concludes the proof of Theorem 1.



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*Source: 6206329-2021-06-30.xml*