# A Hybrid Mean Value Involving Dedekind Sums and the Generalized Kloosterman Sums

Authors: Xiaowei Pan; Xiaoyan Guo
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, we use the mean value theorem of DirichletL-functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.


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## Body

## 1. Introduction

Letq be a natural number and h be an integer coprime to q. The classical Dedekind sums(1)Sh,q=∑a=1qaqahq,where(2)x=x−x−12,ifxis not an integer,0,ifxis an integer,describes the behaviour of the logarithm of the η-function (see [1, 2]) under modular transformations. There are many papers written on their various properties (see the examples in [3–10] and [11]).In particular, Zhang and Liu [12] studied the hybrid mean value problems related to Dedekind sums and Kloosterman sums:(3)Km,n;q=∑a=1q′ema+na¯q,where q≥3 is an integer, ∑a=1q′ denotes the summation over all 1≤a≤q with a,q=1, ey=e2πiy, and a¯ denotes the multiplicative inverse of amodq. They proved the following results:Theorem 1.
Letp be an odd prime, then one has the identity(4)∑a=1p−1∑b=1p−1Ka,1;p2⋅Kb,1;p2⋅Sab¯,p=112⋅p2⋅p−1p−2−12⋅hp2,ifp≡3mod4,112⋅p2⋅p−1p−2,ifp≡1mod4,where hp denotes the class number of the quadratic field Q−p.Theorem 2.
Letp be an odd prime, then one has the asymptotic formula:(5)∑a=1p−1∑b=1p−1Ka,1;p2⋅Kb,1;p2⋅Sab¯,q2=124p5+Op4⋅exp3lnlnplnp,where expy=ey.It is natural that people will ask, for the general Kloosterman sums(6)Km,n,χ;p=∑a=1p−1χaema+na¯p,what will happen? Whether there exists an identity similar to Theorem1? Here, χ denotes any Dirichlet character modp.The main purpose of this paper is to answer these questions. That is, we shall use the mean value theorem of DirichletL-functions and the properties of Gauss sums and Dedekind sums to prove the following.Theorem 3.
Letp be an odd prime with p≡1mod4. Then, for any Dirichlet character χmodp, we have the identity:(7)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=112⋅p2p−1p−2,ifχ−1=1,112⋅p2p−1p−2−p3π2⋅L1,χ2,ifχ−1=−1.Theorem 4.
Letp be an odd prime with p≡3mod4. Then, for any Dirichlet character χmodp, we have the identity:(8)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=112⋅p2p−1p−2−p2⋅hp2,ifχ−1=1,p2p−1p−212−p2⋅hp2−2p3π2⋅L1,χ2,ifχ−1=−1andχ≠χ2,p2p−1p−212+p2⋅p−4⋅hp2,ifχ=χ2,where χ2=∗/p denotes the Legendre symbol and hp denotes the class number of the quadratic field Q−p.It is clear that ifχ=χ0, then Ka,1,χ;p=Ka,1;p. Note that χ0−1=1, from Theorems 3 and 4, we may immediately deduce Theorem1 in [12], so our results are the generalization of [12].


## 2. Several Lemmas

In this section, we shall give several simple lemmas, which are necessary to the proofs of our theorems. Hereafter, we shall use many properties of character sums and Gauss sums, and all of these can be found in reference [13]. First, we have the following.Lemma 1.
Letp>3 be a prime, χ be any fixed Dirichlet character modp. Then, for any nonprincipal character χ1modp with χχ1≠χ0, we have the identity:(9)∑m=1p−1χ1m⋅Km,1,χ;p2=χ¯1−1⋅τ2χ1⋅τχ¯12⋅τχ¯χ1τχ¯χ¯1,where χ0 denotes the principal character modp, τχ denotes the Gauss sums defined as τχ=∑a=1p−1χaea/p, and χ¯ denotes the complex conjugate of χ.Proof.
From the definition of Kloosterman sums and the properties of Gauss sums, we have(10)∑m=1p−1χ1m⋅Km,1,χ;p2=∑m=1p−1χ1m∑a=1p−1∑b=1p−1χaχ¯beam−mb+a¯−b¯p=∑a=1p−1∑b=1p−1χa∑m=1p−1χ1memba−1+b¯⋅a¯−1p=τχ1∑a=1p−1∑b=1p−1χaχ¯1bχ¯1a−1eb¯⋅a¯−1p=τχ1∑a=1p−1χaχ¯1a−1∑b=1p−1χ¯1beb¯⋅a¯−1p=τ2χ1∑a=1p−1χaχ¯1a−1χ¯1a¯−1=χ¯1−1τ2χ1∑a=1p−1χaχ1aχ¯12a−1=χ¯1−1τ2χ1∑a=1p−1χa+1χ1a+1χ¯12a.
On the other hand, from the properties of Gauss sums, we have(11)∑a=1p−1χa+1χ1a+1χ¯12a=1τχ¯χ¯1∑a=1p−1χ¯12a∑b=1p−1χ¯χ¯1beba+1p=1τχ¯χ¯1∑b=1p−1χ¯χ¯1bebp∑a=1p−1χ¯12aebap=τχ¯12τχ¯χ¯1∑b=1p−1χ¯bχ1bebp=τχ¯12⋅τχ¯χ1τχ¯χ¯1.
Combining (10) and (11), we may immediately deduce the identity:(12)∑m=1p−1χ1m⋅Km,1,χ;p2=χ¯1−1⋅τ2χ1⋅τχ¯12⋅τχ¯χ1τχ¯χ¯1.
This proves Lemma1.Lemma 2.
Letp be a prime with p≡1mod4 and χ be any odd character modp. Then, we have the identity:(13)∑m=1p−1χ¯m⋅Km,1,χ;p2=τ2χ¯.Proof.
Sincep≡1mod4 and χ is an odd character modp, we know χ is not the Legendre symbol and χ2≠χ0. Note that χ−1=−1, from (10), we have(14)∑m=1p−1χ¯m⋅Km,1,χ;p2=τ2χ¯∑a=1p−1χaχa−1χa¯−1=χ−1τ2χ¯∑a=1p−1χ2a−1=−τ2χ¯∑a=0p−1χ2a−1=τ2χ¯.
This proves Lemma2.Lemma 3.
Letp be a prime with p≡3mod4 and χ2 be the Legendre symbol. Then, we have the identity:(15)∑m=1p−1χ2mKm,1,χ2;p2=−p−2⋅τ2χ2.Proof.
Note thatχ2=χ¯2, χ22=χ0, and χ2−1=−1, and from the definition of Km,1,χ2;p and the properties of Gauss sums, we have(16)∑m=1p−1χ2mKm,1,χ2;p2=∑a=1p−1∑b=1p−1χ2ab∑m=1p−1χ2mema−b+a¯−b¯p=τχ2∑a=1p−1χ2a∑b=1p−1χ2ba−1eb¯a¯−1p=τ2χ2∑a=1p−1χ2aχ2a−1χ2a¯−1=−p−2⋅τ2χ2.
This proves Lemma3.Lemma 4.
Letq>2 be an integer, then for any integer a with a,q=1, we have the identity:(17)Sa,q=1π2q∑d|qd2ϕd∑χmoddχ−1=−1χaL1,χ2,where L1,χ denotes the Dirichlet L-function corresponding to the character χmodd.Proof.
See Lemma2 of [7].


## 3. Proof of the Theorems

In this section, we will complete the proof of our theorems. First we prove Theorem3. From Lemma 4 and the definition of Sa,p, we have(18)Sa,p=1π2⋅pp−1⋅∑χmoddχ−1=−1χaL1,χ2,and (with a=1)(19)∑χmoddχ−1=−1L1,χ2=π2⋅p−1p⋅S1,p=π2⋅p−1p⋅∑a=1p−1ap−122=π212⋅p−12p−2p2.Sincep≡1mod4, we know the Legendre symbol ∗/p=χ2 is an even character modp. Note that, for any nonprincipal character χmodp, τχ=p. So, if χ is an even character modp, then from Lemma 1, (18), and (19), we have(20)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=p⋅π−2p−1∑χ1moddχ1−1=−1∑a=1p−1χ1aKa,1,χ;p22⋅L1,χ12=p⋅π−2p−1∑χ1moddχ1−1=−1τ2χ1⋅τχ¯12⋅τχ¯χ1τχ¯χ¯12⋅L1,χ12=p4⋅π−2p−1∑χ1moddχ1−1=−1L1,χ12=112⋅p2p−1p−2.Ifχ is an odd character modp, then note that the identity:(21)∑χ1modpχ1−1=−1fχ1=∑χ1modpχ1−1=−1χ1χ≠χ0fχ1+fχ¯,from (18), (19), Lemmas 1 and 2, and the method of proving (20), we have(22)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=p⋅π−2p−1∑χ1modpχ1−1=−1∑a=1p−1χ1aKa,1,χ;p22⋅L1,χ12=p⋅π−2p−1∑χ1modpχ1−1=−1χ1χ≠χ0∑a=1p−1χ1aKa,1,χ;p22⋅L1,χ12+p⋅π−2p−1∑χ1modpχ1−1=−1χ1χ=χ0∑a=1p−1χ1aKa,1,χ;p22⋅L1,χ12=p⋅π−2p−1∑χ1modpχ1−1=−1χ1χ≠χ0τ2χ1⋅τχ¯12⋅τχ¯χ1τχ¯χ¯12⋅L1,χ12+p⋅π−2p−1τ2χ¯2⋅L1,χ¯2=p4⋅π−2p−1∑χ1modpχ1−1=−1L1,χ12−p4⋅π−2p−1L1,χ2+p3⋅π−2p−1L1,χ2=112⋅p2p−1p−2−p3π2⋅L1,χ2,where ∑χ1modpχ1−1=−1χ1χ≠χ0 denotes the summation over all odd characters χ1 with χ1≠χ¯.Combining (20) and (22), we may immediately deduce the identity:(23)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=112⋅p2p−1p−2,ifχ−1=1,112⋅p2p−1p−2−p3π2⋅L1,χ2,ifχ−1=−1.This proves Theorem3.Now, we prove Theorem4. Since p≡3mod4, we know the Legendre symbol χ2 is an odd character modp and τχ¯22=1. If χ is an even character modp, then note that L1,χ2=π⋅hp/p (see reference [14]), from (18), (19), Lemma 1 and the properties of Gauss sums, we have(24)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=p⋅π−2p−1∑χ1modpχ1−1=−1τ2χ1⋅τχ¯12⋅τχ¯χ1τχ¯χ¯12⋅L1,χ12=p⋅π−2p−1∑χ1modpχ1−1=−1χ1≠χ2τ2χ1⋅τχ¯12⋅τχ¯χ1τχ¯χ¯12⋅L1,χ12+p3⋅π−2p−1⋅L1,χ22=p4⋅π−2p−1∑χ1modpχ1−1=−1L1,χ12+p3⋅π−2p−1⋅L1,χ22−p4⋅π−2p−1⋅L1,χ22=112⋅p2p−1p−2−p2⋅hp2,where hp denotes the class number of the quadratic field Q−p.Ifχ is an odd nonreal character modp, then from (18), (19), Lemmas 1 and 2, and the method of proving (22) and (24), we have(25)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=p⋅π−2p−1∑χ1modpχ1−1=−1∑a=1p−1χ1aKa,1,χ;p22⋅L1,χ12=p⋅π−2p−1∑χ1modpχ1−1=−1χ1χ≠χ0τ2χ1⋅τχ¯12⋅τχ¯χ1τχ¯χ¯12⋅L1,χ12+p⋅π−2p−1∑a=1p−1χ¯aKa,1,χ;p22⋅L1,χ¯2=p4⋅π−2p−1∑χ1modpχ1−1=−1L1,χ12+p3⋅π−2p−1⋅L1,χ¯2+p3⋅π−2p−1⋅L1,χ2+p3⋅π−2p−1⋅L1,χ22−2p4⋅π−2p−1⋅L1,χ2−p4⋅π−2p−1⋅L1,χ22=112⋅p2p−1p−2−p2⋅hp2−2p3π2⋅L1,χ2.Ifχ=χ2 is the Legendre symbol, then from (18), (19), Lemmas 1 and 3, and the method of proving (25), we have(26)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=p⋅π−2p−1∑χ1modpχ1−1=−1∑a=1p−1χ1aKa,1,χ;p22⋅L1,χ12=p⋅π−2p−1∑χ1modpχ1−1=−1χ1≠χ2τ2χ1⋅τχ¯12⋅τχ¯2χ1τχ¯2χ¯12⋅L1,χ12+p⋅π−2p−1∑a=1p−1χ¯2aKa,1,χ2;p22⋅L1,χ¯22=p4⋅π−2p−1∑χ1modpχ1−1=−1L1,χ12−p4⋅π−2p−1⋅L1,χ22+p3⋅π−2p−1⋅p−22⋅L1,χ22=p2p−1p−212+p2⋅p−4⋅hp2.Combining (24), (25), and (26), we can deduce the identity:(27)∑a=1p−1∑b=1p−1Ka,1,χ;p2⋅Kb,1,χ;p2⋅Sab¯,p=112⋅p2p−1p−2−p2⋅hp2,ifχ−1=1,p2p−1p−212−p2⋅hp2−2p3π2⋅L1,χ2,ifχ−1=−1andχ≠χ2,p2p−1p−212+p2⋅p−4⋅hp2,ifχ=χ2.This completes the proof of Theorem4.



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*Source: 5516217-2021-03-30.xml*