# On the Hybrid Power Mean Involving the Character Sums and Dedekind Sums

Authors: Xiaoling Xu
Journal: Journal of Mathematics
 (2021)

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## Abstract

The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.


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## Body

## 1. Introduction

We all know that the classical Dedekind sumsSh,q is defined (see [1]) as(1)Sh,q=∑a=1qaqahq,where q≥2 is a positive integer, h is any integer prime to q, and(2)x=x−x−12,ifxis not an integer,0,ifxis an integer.This sum describes the behaviour of the logarithm of the eta function under modular transformations, see [1, 2], for related references. Because of the importance of this sum in analytic number theory, many scholars have studied its various properties and obtained a series of important results. Perhaps, the most important property of Sh,q is its reciprocity theorem (see [3]). That is, for any positive integers h and q with h,q=1, one has the identity(3)Sh,q+Sq,h=h2+k2+112hq−14.Some other papers related to Dedekind sums can be found in [4–6], and we do not want to list them all here.On the contrary, we also introduce another character sums analogous to Kloosterman sums as follows. For any integerq≥3, let χ be a Dirichlet character modq. For any positive integer k and integer h, we define(4)Gk,h,χ;q=∑a1=1q′∑a2=1q′…∑ak=1qχa1+a2+…+ak+ha1⋅a2…ak¯,where ∑a=1q′ denotes the summation over all 1≤a≤q such that a,q=1 and a¯ denotes the inverse of a. That is, a⋅a¯≡1modq.About the properties ofGk,h,χ;q, some people had studied it and obtained some important results. For example, from the very special case of Weil’s work [7] one can obtain the estimate(5)∑a=1p−1χa+ma¯≤2p,where p is a prime and m is any integer. Some related important works can also be found in [7–11].In this paper, we consider the computational problem of the hybrid power mean involving the Dedekind sumsSh,q and Gk,h,χ;q. That is,(6)∑h=1q′Gk,h,χ;q⋅Sh,q.However, for this hybrid power mean, it seems that none has studied it yet; at least, we have not seen any related results before. The problem is interesting because it is closely related to Dirichlet L-functions. In fact, for some special positive integersk, we can give an exact computational formula for (6). The main work of this paper is to reveal this point. That is, we shall use the elementary and analytic methods, and the properties of character sums to prove the following two conclusions.Theorem 1.
Letp be an odd prime with p≡3mod4, and χ2=∗/p denotes Legendre’s symbol modp. Then, for any positive integer k with 2k+1,p−1=1, we have the identity(7)∑h=1p−1G2k,h,χ2;p⋅Sh,p=−1k⋅pk⋅hp2,where hp denotes the class number of the quadratic field Q−p.Theorem 2.
Letp be an odd prime with p≡1mod8 and k be any positive integer with k,p−1=1. Then, we have the identity(8)∑h=1p−1G2k−1,h,χ2;p⋅Sh,p=0.
Ifp be an odd prime with p≡5mod8, then we have the identity(9)∑h=1p−1G2k−1,h,χ2;p⋅Sh,p=1π2⋅pk+1/2⋅L1,χ42⋅∑h=0k/2−1h⋅kk−h⋅k−hh⋅2αk−2h⋅ph,where χ4 denotes the fourth-order character modp, α=∑a=1p−1/2a+a¯/p is an integer, and Ls,χ denotes the Dirichlet L-function corresponding to χ.
Takingk=1 in Theorem 1 and Theorem 2, then we have the following.Corollary 1.
Letp be an odd prime with p≡3mod4 and 3,p−1=1; then, we have the identity(10)∑h=1p−1G2,h,χ2;p⋅Sh,p=−p⋅hp2.Corollary 2.
Letp be an odd prime with p≡5mod8; then, we have(11)∑h=1p−1G1,h,χ2;p⋅Sh,p=2απ2⋅p⋅L1,χ42.Notes: Obviously, in a sense, Corollary1 gives us efficient methods to compute the class number hp that can be done on a computer.It is easy to prove that ifp≡3mod4, then, for any positive integer k, we have(12)∑h=1p−1G2k−1,h,χ2;p⋅Sh,p=0.Ifp≡1mod4, then, for any positive integer k, we also have(13)∑h=1p−1G2k,h,χ2;p⋅Sh,p=0.For general composite numberq>3, whether there exists an exact computational formula for (6) will be our further research problem.


## 2. Several Lemmas

In this section, we shall give several simple lemmas, and they are necessary in the proofs of our theorems. First, we have the following.Lemma 1.
Letp>3 be a prime, and λ and χ are two nonprincipal characters modp with χ−1=−1. Then, for any positive integer k, we have the identity(14)∑m=1p−1χmGk,m,λ;p=p−1⋅τk+1χτλ¯,ifλ=χ¯k+1;0,otherwise,where τχ denotes the classical Gauss sums.Proof.
For any integern and nonprincipal character χmodp, from the properties of Gauss sums τχ (see Theorem 8.20 in [12]), we have(15)χn=1τχ¯∑a=1p−1χ¯aenap.
Using (15) and the properties of the reduced residue system modp, we have(16)Gk,m,λ;p=∑a1=1p−1∑a2=1p−1…∑ak=1p−1λa1+a2+…+ak+ma1a2,…,ak¯=1τλ¯∑b=1p−1λ¯b∑a1=1p−1∑a2=1p−1…∑ak=1p−1eba1+a2+…+ak+bma1,…,ak¯p=1τλ¯∑b=1p−1λ¯b∑a1=1p−1…∑ak=1p−1ea1+…+ak+bk+1ma1,…,ak¯p.
So, with the repeated use of (15) in (16), we have(17)∑m=1p−1χmGk,m,λ;p=1τλ¯∑b=1p−1λ¯b∑a1=1p−1…∑ak=1p−1∑m=1p−1χmea1+…+ak+bk+1ma1…ak¯p=τχτλ¯∑b=1p−1λ¯bχ¯k+1b∑a1=1p−1…∑ak=1p−1χa1…akea1+…+akp=τk+1χτλ¯∑b=1p−1λ¯bχ¯k+1b=p−1⋅τk+1χτλ¯,ifλ=χ¯k+1,0,otherwise.
This proves Lemma1.Lemma 2.
Letq>2 be an integer; then, for any integer a with a,q=1, we have the identity(18)Sa,q=1π2q∑d/qd2ϕd∑χmoddχ−1=−1χaL1,χ2,where L1,χ denotes the Dirichlet L-function corresponding to character χmodd.Proof.
See Lemma2 of [6].Lemma 3.
Ifp is a prime with p≡1mod4 and ψ is any fourth-order character modp, then we have the identity(19)τ2ψ+τ2ψ¯=2p⋅α,where α=∑a=1p−1/2a+a¯/p is an integer.Proof.
See Lemma 2.2 of [13] or Lemma 3 of [14].Lemma 4.
Ifp is a prime with p≡5mod8 and ψ is any fourth-order character modp, then, for any positive integer k, we have the identity(20)τ2kψ+τ2kψ¯=∑h=0k/2−1h⋅kk−h⋅k−hh⋅2αk−2h⋅p1/2k+2h.Proof.
First, for all nonnegative integersu and real numbers X and Y, we have the identity(21)Xu+Yu=∑h=0u/2−1h⋅uu−hu−hh⋅X+Yu−2h⋅XYh,where x denotes the greatest integer ≤x. This formula is obtained because of Waring [15]. It can also be found in [16].
Note that ifp≡5mod8, then, for any fourth-order character ψmodp, we have τψ¯=ψ¯−1⋅τψ¯=−τψ¯. So, τψ¯⋅τψ=−p. Thus, taking X=τ2ψ and Y=τ2ψ¯, from (4) and Lemma 3, we have(22)τ2kψ+τ2kψ¯=∑h=0k/2−1h⋅kk−hk−hh⋅τ2ψ+τ2ψ¯k−2h⋅p2h=∑h=0k/2−1h⋅kk−hk−hh⋅2p⋅αk−2h⋅p2h=∑h=0k/2−1h⋅kk−h⋅k−hh⋅2αk−2h⋅p1/2k+2h.
This proves Lemma4.


## 3. Proofs of the Theorems

In this section, we shall complete the proofs of our theorems. First, we prove Theorem1. From Lemma 2, we have(23)Sa,p=1π2⋅pp−1⋅∑χmodpχ−1=−1χaL1,χ2.Ifp is a prime with p≡3mod4, then, for any positive integer k with 2k+1,p−1=1, let 2k+1¯⋅2k+1≡1modp−1; then, 2k+1¯ must be an odd number. If characters λ and χ satisfy λ=χ¯2k+1 with χ−1=−1, then χ¯=λ2k+1¯. If λ=∗/p=χ2 is Legendre’s symbol modp, then we have χ¯=χ22k+1¯=χ2. Note that if p≡3mod4, then χ2−1=−1, τχ2=ip and L1,χ2=πhp/p. So, from (23) and Lemma 1, we have(24)∑h=1p−1G2k,h,χ2;p⋅Sh,p=1π2⋅pp−1⋅∑χmodpχ−1=−1∑h=1p−1χh⋅G2k,h,χ2;p⋅L1,χ2=1π2⋅p⋅∑χmodpχ=χ2τ2k+1χi⋅p⋅L1,χ2=1π2⋅p⋅i⋅p2k⋅π2p⋅hp2=−1k⋅pk⋅hp2.This proves Theorem1.Now, we prove Theorem2. Let p be a prime with p≡5mod8; then, τχ2=p. For any four-order character χmodp, we have χ−1=−1. So, for any positive integer k with k,p−1=1, note that k¯ is an odd number; if χ2=χ2k, then χ2=χ2. In this time, χ=χ4 must be a fourth-order character modp and L1,χ4=L1,χ¯4; from (23), Lemma 1, and Lemma 4, we have(25)∑h=1p−1G2k−1,h,χ2;p⋅Sh,p=1π2⋅pp−1⋅∑χmodpχ−1=−1∑h=1p−1χh⋅G2k−1,h,χ2;p⋅L1,χ2=1π2⋅p⋅∑χmodpχ2=χ2τ2kχp⋅L1,χ2=1π2⋅p⋅τ2kχ4p⋅L1,χ42+τ2kχ¯4p⋅L1,χ¯42=1π2⋅p⋅τ2kχ4+τ2kχ¯4⋅L1,χ42=1π2⋅p⋅∑h=0k/2−1h⋅kk−h⋅k−hh⋅2αk−2h⋅p1/2k+2h⋅L1,χ42=1π2⋅pk+1/2⋅∑h=0k/2−1h⋅kk−h⋅k−hh⋅2αk−2h⋅ph⋅L1,χ42.Ifp≡1mod8, then χ4−1=1, so in this time, we have(26)∑h=1p−1G2k−1,h,χ2;p⋅Sh,p=1π2⋅pp−1⋅∑χmodpχ−1=−1∑h=1p−1χh⋅G2k−1,h,χ2;p⋅L1,χ2=1π2⋅p⋅∑χmodpχ2=χ2τ2kχp⋅L1,χ2=0.Combining (25) and (26), we may immediately deduce Theorem 2.



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