A new reciprocity formula of Dedekind sums and its applications

Zhuoyu Chen; Wenpeng Zhang; Zhuoyu Chen; Wenpeng Zhang

doi:10.3934/math.2024626

AIMS Mathematics

2024, Volume 9, Issue 5: 12814-12824. doi: 10.3934/math.2024626

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Research article Special Issues

A new reciprocity formula of Dedekind sums and its applications

Zhuoyu Chen ¹,
Wenpeng Zhang ^{2
,
,}

1.
School of Freshmen, Xi'an Technological University, Xi'an 710021, China
2.
School of Science, Xi'an Technological University, Xi'an 710021, China
^†Dedicated to Professor Taekyun Kim on the occasion of his sixtieth birthday

Received: 27 February 2024 Revised: 21 March 2024 Accepted: 26 March 2024 Published: 03 April 2024
MSC : 11F20, 11M20

Our main purpose of this article was using the analytic methods and the properties of Dirichlet $L$ -functions to study the properties of Dedekind sums and give a new reciprocity formula for it. As its applications, some exact calculating formula for one kind mean square value of Dirichlet $L$ -fuctions with the weight of the character sums were obtained.

Keywords:

Citation: Zhuoyu Chen, Wenpeng Zhang. A new reciprocity formula of Dedekind sums and its applications[J]. AIMS Mathematics, 2024, 9(5): 12814-12824. doi: 10.3934/math.2024626

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Abstract

1. Introduction

In order to describe the results of this paper, we first introduce the definition of the Dedekind sums $S(r, q)$ . For any integers $q\geq 2$ and $r$ , the classical Dedekind sums $S(r, q)$ is defined as follows (see ^[1]):

$S(r, q) = \sum\limits^q_{c = 1}\left(\left(\frac{c}{q}\right)\right) \left(\left(\frac{rc}{q}\right)\right),$

where $((u))$ is usually defined as

$((u)) = \left\{ \begin{array}{ll} u - [u] - \frac{1}{2}, & \text{ if}~~ u ~~\text{ is not an integer;}\\0, & \text{ if }~~ u ~~\text{is an integer.} \end{array}\right.$

Usually, we know that $S(r, q)$ describes the behavior of the logarithm for the eta-function (see ^[2,3]) under modular transformations. Because of the importance of $S(r, q)$ in analytic number theory, many authors have studied the various arithmetical properties of $S(r, q)$ and obtained a series of meaningful results (some of them can be found in ^{[4,5,6,7,8,9,10,11,12,13,14,15,16]}). To avoid complexity, we do not want to list them one by one. However, it is worth mentioning that Girstmair acquired an interesting result in ^[17], and it should be noted that perhaps the most important property of $S(r, q)$ is its reciprocity theorem (see^[1,4,7]). That is, for any positive integers $u$ and $v$ with $(u, v) = 1$ , one has the following identity

$\begin{eqnarray} S(u, v)+ S(v, u) = \frac{u^2+v^2+1}{12uv}-\frac{1}{4}. \end{eqnarray}$

(1.1)

Obviously, this formula is not only looks very beautiful, but also reveals the profound properties between $S(u, v)$ and $S(v, u)$ . Rademacher and Grosswald ^[3] also obtained a three-term formula similar to (1.1). Besides, there are many properties of Dedekind sums that are worth studying, and many scholars have achieved rich results. In particular, some of the new papers related to Dedekind sums can also be found in the references ^{[18,19,20,21,22]}.

Our main purpose of this paper is using the analytic methods and the properties of Dirichlet $L$ -functions to study the properties of $S(r, q)$ and give a new reciprocity formula for it, which is Lemma 3 in the paper. As its applications, we deduced several new calculating formula for the mean square value of Dirichlet $L$ -functions with the weight of the character sums. In other words, we have the following three results:

Theorem 1. For any positive integer $q > 1$ and $(q, 6) = 1$ , we have the following identities

$\begin{eqnarray*} \mathop{\sum\limits_{\chi ~ \text{mod}~ q}}_{\chi(-1) = -1} \chi(3)\cdot\overline{\chi}(2)\cdot |L(1, \chi)|^2 = \frac{\pi^2}{18}\cdot \frac{\phi^2(q)}{q^2}\cdot \left[ \frac{q}{4}\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-\frac{9}{2}+ \left(\frac{q}{3}\right)\cdot \prod\limits_{p\mid q}\frac{p-\left(\frac{p}{3}\right)}{p-1}\right], \end{eqnarray*}$

where $\mathop{\sum\limits_{\chi \bmod q}}_{\chi(-1) = -1}$ denotes the summation over all odd characters modulo $q$ and $\prod_{p\mid q}$ represents the product over all distinct prime divisors of $q$ , $L(s, \chi)$ is the Dirichlet $L$ -function corresponding to character $\chi\bmod q$ , $\phi(q)$ is the Euler function and $\left(\frac{*}{3}\right)$ is the Legendre symbol modulo $3$ .

Theorem 2. For any positive integer $q > 1$ with $(q, 6) = 1$ , we also have the identity

$\begin{eqnarray*} \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = 1} |L(1, \chi\lambda_3)|^2 = \frac{\pi^2}{27}\cdot \frac{\phi^2(q)}{q^2}\left[2q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-\left(\frac{q}{3}\right)\cdot \prod\limits_{p\mid q}\frac{p-\left(\frac{p}{3}\right)}{p-1}\right], \end{eqnarray*}$

where $\sum\limits_{\substack{\chi \bmod h \\ \chi(-1)=-1}}$ denotes the summation over all even characters modulo $q$ , and $\lambda_3 = \left(\frac{*}{3}\right)$ denotes the Legendre symbol modulo $3$ .

Theorem 3. For any positive integer $q > 1$ with $(q, 6) = 1$ , we can obtain the identity

$\begin{eqnarray*} \mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = 1}\lambda(2)\cdot|L(1, \lambda\psi_3)|^2 = -\frac{\pi^2}{27}\cdot \frac{\phi^2(q)}{q^2}\left[ q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-2\left(\frac{q}{3}\right)\cdot\prod\limits_{p\mid q}\frac{p-\left(\frac{p}{3}\right)}{p-1} \right], \end{eqnarray*}$

where

$\psi_3 = \left(\frac{*}{3}\right)$

denotes the Legendre symbol modulo $3$ .

To illustrate the interest of this article, here we provide several numerical examples to justify our obtained results. If we take $q = 5$ in Theorems 1 and 2, we could obtain the following two corollaries:

Corollary 1. Let $\psi$ be any non-real character modulo $5$ . We obtain the identity

$\left|L(1, \psi)\right| = \frac{\sqrt{2}}{5}\cdot \pi.$

Corollary 2. Let $\lambda = \left(\frac{*}{3}\right)\cdot\left(\frac{*}{5}\right)$ modulo $15$ . We have the identity

$\left|L(1, \lambda)\right| = \frac{2\pi}{\sqrt{15}}.$

If we take $q = 7$ in Theorem 2 and $q = 35$ , Theorems 2 and 3, we can also have the following corollaries:

Corollary 3. Let $\chi$ be any non-real even character modulo $7$ . Then we have the identity

$\left|L(1, \chi)\right| = \frac{2\pi}{\sqrt{21}}.$

Corollary 4. Let $\psi_3$ denote the Legendre symbol modulo $3$ . Then we have

$\begin{eqnarray*} \mathop{\sum\limits_{\lambda ~ \text{mod}~ 35}}_{\lambda(-1) = 1}\lambda(2)\cdot|L(1, \lambda\psi_3)|^2 = -\frac{9792}{11025}\cdot \pi^2. \end{eqnarray*}$

Corollary 5. Let $\psi_3$ denote the Legendre symbol modulo $3$ . Then we have

$\begin{eqnarray*} \mathop{\sum\limits_{\chi ~ \text{mod}~ 35}}_{\chi(-1) = 1}|L(1, \chi\psi_3)|^2 = \frac{416}{245}\cdot \pi^2. \end{eqnarray*}$

Some notes: It is clear that by replacing $\chi(3)$ with $\chi(5)$ or $\chi(7)$ in Theorem 1 we can also get some similar results, but the situation is more complicated and we do not list them.

In addition, Theorems 2 and 3 show two interesting results. In fact, for the mean square value of Dirichlet $L$ -functions with the even characters at point $s = 1$ , there are so far only various asymptotic formulas, without any exact identities. For Theorems 2 and 3, we obtained two exact calculating formulae for them just by turning all characters $\chi$ with $\chi(-1) = 1$ into $\chi\lambda_3$ with $\chi\lambda_3(-1) = -1$ .

2. Several lemmas

In this section, we will give three simple lemmas that are necessary in the proofs of three theorems. Hereinafter, we shall combine some knowledge of analytic number theory and the properties of the Dirichlet $L$ -functions and Dedekind sums, which can be found in references ^[1,23,24], so we will not repeat them here. First, we have the following:

Lemma 1. Let $q > 2$ be an integer. Then for any integer $v$ with $(v, q) = 1$ , we have the following equation:

$S(v, q) = \frac{1}{\pi^2 q}\sum\limits_{h|q}\frac{h^2}{\phi(h)}\sum\limits_{\substack{\chi \bmod h \\ \chi(-1)=-1}} \chi(v)|L(1, \chi)|^2,$

where $L(s, \chi)$ denotes the Dirichlet $L$ -function corresponding to $\chi \bmod h$ .

Proof. See [6,Lemma 2]. □

Lemma 2. Let $q$ be a positive odd number. For any positive odd number $r$ with $(r, q) = 1$ , we can calculate the identity

$S\left(r, 2q\right) = 3\cdot S(r, q)- S\left(2r, q\right)-S\left(\overline{2}r, q\right),$

where $2\cdot \overline{2}\equiv 1\bmod q$ .

Proof. Let $\chi_0$ denote the principal character modulo $2$ . Note that $2\nmid q$ and for any $h\mid q$ , we know that

$\phi(2h) = \phi(h)$

and

$(r, 2q) = 1.$

According to the Lemma 1, we can obtain

$\begin{align} \begin{split} S\left(r, 2q\right) & = \frac{1}{2\pi^2 q}\sum\limits_{h|2q}\frac{h^2}{\phi(h)}\mathop{\sum\limits_{\chi ~ \text{mod}~ h}}_{\chi(-1) = -1} \chi(r)|L(1, \chi)|^2\\ & = \frac{1}{2\pi^2 q}\sum\limits_{h|q}\frac{h^2}{\phi(h)}\mathop{\sum\limits_{\chi ~ \text{mod}~ h}}_{\chi(-1) = -1} \chi(r)|L(1, \chi)|^2+ \frac{1}{2\pi^2 q}\sum\limits_{h|q}\frac{(2h)^2}{\phi(2h)}\mathop{\sum\limits_{\chi ~ \text{mod}~ 2h}}_{\chi(-1) = -1} \chi(r)|L(1, \chi)|^2\\ & = \frac{1}{2}\cdot S(r, q)+ \frac{2}{\pi^2 q}\sum\limits_{h|q}\frac{h^2}{\phi(h)}\mathop{\sum\limits_{\chi ~ \text{mod}~ h}}_{\chi(-1) = -1} \chi(r)|L(1, \chi\chi_0)|^2.\end{split} \end{align}$

(2.1)

For any non-principal character $\chi\bmod h$ with $h\mid q$ , using the properties of Dirichlet $L$ -functions and the Euler product formula (see [23,Theorem 11.6]), we can calculate that

$\begin{align} \begin{split} |L(1, \chi\chi_0)|^2 & = \left|\prod\limits_{p}\left(1-\frac{\chi(p)\chi_0(p)}{p}\right)^{-1} \right|^2\\ & = \left|1-\frac{\chi(2)}{2}\right|^2\cdot \left|\prod\limits_{p}\left(1-\frac{\chi(p)}{p}\right)^{-1} \right|^2\\ & = \left(\frac{5}{4}-\frac{\chi(2)}{2}-\frac{\overline{\chi}(2)}{2}\right)\cdot \left|L(1, \chi)\right|^2.\end{split} \end{align}$

(2.2)

Combining (2.1), (2.2) and Lemma 1, we have the identity

$\begin{align*} S\left(r, 2q\right) = &\frac{1}{2}\cdot S(r, q)+ \frac{2}{\pi^2 q}\sum\limits_{h|q}\frac{h^2}{\phi(h)}\mathop{\sum\limits_{\chi ~ \text{mod}~ h}}_{\chi(-1) = -1} \chi(r)\left(\frac{5}{4}-\frac{\chi(2)}{2}-\frac{\overline{\chi}(2)}{2}\right)\cdot \left|L(1, \chi)\right|^2\nonumber\\ = &\frac{1}{2}\cdot S(r, q)+ \frac{5}{2}\cdot S(r, q)- \frac{1}{\pi^2 q}\sum\limits_{h|q}\frac{h^2}{\phi(h)}\mathop{\sum\limits_{\chi ~ \text{mod}~ h}}_{\chi(-1) = -1} \chi(2r)|L(1, \chi)|^2\nonumber\\ &-\frac{1}{\pi^2 q}\sum\limits_{h|q}\frac{h^2}{\phi(h)}\mathop{\sum\limits_{\chi ~ \text{mod}~ h}}_{\chi(-1) = -1} \chi\left(\overline{2}r\right)|L(1, \chi)|^2\nonumber\\ = &3\cdot S(r, q)- S\left(2r, q\right)-S\left(\overline{2}r, q\right). \end{align*}$

This completes the proof of Lemma 2. □

Lemma 3. Let $h$ and $q$ be two positive odd numbers with $(h, q) = 1$ . Then we have the reciprocity formula

$\begin{eqnarray*} S\left(\overline{2}q, h\right)+ S\left(\overline{2}h, q\right) = \frac{h^2+q^2+4}{24hq}-\frac{1}{4}, \end{eqnarray*}$

where $\overline{2}$ in $S\left(\overline{2}h, q\right)$ and $S\left(\overline{2}q, h\right)$ are $\frac{q+1}{2}$ and $\frac{h+1}{2}$ , respectively.

Proof. For any positive odd numbers $h$ and $q$ with $(h, q) = 1$ , according to the Lemma 2, we can obtain

$\begin{eqnarray} S\left(h, 2q\right) = 3\cdot S(h, q)- S\left(2h, q\right)-S\left(\overline{2}h, q\right) \end{eqnarray}$

(2.3)

and

$\begin{eqnarray} S\left(q, 2h\right) = 3\cdot S(q, h)- S\left(2q, h\right)-S\left(\overline{2}q, h\right). \end{eqnarray}$

(2.4)

Applying (2.3), (2.4) and the reciprocity formula (1.1), we have the following identity:

$\begin{eqnarray*} S\left(h, 2q\right)+S\left(2q, h\right)+S\left(q, 2h\right)+ S\left(2h, q\right) = 3\cdot S(h, q) + 3\cdot S(q, h)- S\left(\overline{2}q, h\right)-S\left(\overline{2}h, q\right) \end{eqnarray*}$

$\begin{eqnarray*} \frac{4q^2+h^2+1}{24hq}-\frac{1}{4}+\frac{4h^2+q^2+1}{24qh}-\frac{1}{4} = 3\cdot \left(\frac{h^2+q^2+1}{12hq}-\frac{1}{4}\right)- S\left(\overline{2}q, h\right)-S\left(\overline{2}h, q\right) \end{eqnarray*}$

$\begin{eqnarray*} S\left(\overline{2}q, h\right)+S\left(\overline{2}h, q\right) = \frac{h^2+q^2+4}{24hq}-\frac{1}{4}. \end{eqnarray*}$

This proves Lemma 3. □

It is clear that our Lemma 3 gave a new reciprocity formula for Dedekind sums.

3. Proofs of the theorems

In this section, we use the three simple lemmas in Section 2 and the reciprocity formula (1.1) to prove our three results. We will first prove Theorem 1.

Proof. Taking $h = 3$ in Lemma 3 and because of $2\cdot 2\equiv 1\bmod 3$ , if $q = 6k+1$ , from the definition of $S(h, q)$ , we can obtain

$\begin{eqnarray} S\left(\overline{2}q, 3\right) = S\left(2q, 3\right) = S(2, 3) = -S(1, 3) = -\frac{1}{18}. \end{eqnarray}$

(3.1)

If $q = 6k-1$ , from the definition of $S(h, q)$ , we can also obtain

$\begin{eqnarray} S\left(\overline{2}q, 3\right) = S\left(2q, 3\right) = S(-2, 3) = S(1, 3) = \frac{1}{18}. \end{eqnarray}$

(3.2)

Combining (3.1) and (3.2), for any odd positive integer $q$ with $(q, 3) = 1$ , we have

$\begin{eqnarray} S\left(\overline{2}q, 3\right) = S\left(2q, 3\right) = -\frac{\left(\frac{q}{3}\right)}{18}. \end{eqnarray}$

(3.3)

Note that $S(r+hq, q) = S(r, q)$ , from (3.3) and Lemma 3 we have

$\begin{eqnarray} S\left(\overline{2}\cdot3, q\right) = S\left(\frac{3q+3}{2}, q\right) = S\left(\frac{q+3}{2}, q\right) = \frac{q^2+13}{72q}-\frac{1}{4}+\frac{\left(\frac{q}{3}\right)}{18}. \end{eqnarray}$

(3.4)

According to the formula (3.1)–(3.3) and the properties of the Möbius function, we calculate

$\begin{align} \begin{split} \sum\limits_{d\mid q} \mu(d)\cdot \frac{q}{d}\cdot S\left(\frac{q}{d}, 3\right) & = \sum\limits_{d\mid q} \mu(d)\cdot \frac{q}{d}\cdot \frac{\left(\frac{q/d}{3}\right)}{18} = \frac{q}{18}\cdot\left(\frac{q}{3}\right)\cdot\sum\limits_{d\mid q} \frac{\mu(d)\cdot \left(\frac{d}{3}\right)}{d}\\ & = \phi(q)\cdot \frac{\left(\frac{q}{3}\right)}{18}\cdot \prod\limits_{p\mid q}\frac{p-\left(\frac{p}{3}\right)}{p-1}, \end{split} \end{align}$

(3.5)

where $\mu(n)$ denotes the Möbius function.

From the formula (3.4) and Lemma 1 we have

$\begin{eqnarray} \frac{q}{72}-\frac{1}{4}+\frac{13}{72q}+\frac{\left(\frac{q}{3}\right)}{18} = \frac{1}{\pi^2 q}\sum\limits_{k|q}\frac{k^2}{\phi(k)}\mathop{\sum\limits_{\chi ~ \text{mod}~ k}}_{\chi(-1) = -1} \chi(3)\cdot\overline{\chi}(2)\cdot |L(1, \chi)|^2. \end{eqnarray}$

(3.6)

Note that (3.5), applying Möbius inversion formula (see [23,Theorem 2.9]) for (3.6) we have

$\begin{align*} \mathop{\sum\limits_{\chi ~ \text{mod}~ q}}_{\chi(-1) = -1} \chi(3)\cdot\overline{\chi}(2)\cdot |L(1, \chi)|^2 & = \frac{\pi^2\cdot \phi(q)}{q^2}\cdot \sum\limits_{d\mid q}\mu(d)\left(\frac{q^2}{72d^2}-\frac{1}{4}\cdot \frac{q}{d}+ \frac{\left(\frac{q/d}{3}\right)}{18}+\frac{13}{72}\right)\nonumber\\ & = \pi^2\cdot \frac{\phi^2(q)}{q^2}\cdot \left[\frac{1}{72}\cdot q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-\frac{1}{4}+ \frac{\left(\frac{q}{3}\right)}{18}\cdot \prod\limits_{p\mid q}\frac{p-\left(\frac{p}{3}\right)}{p-1}\right]. \end{align*}$

The proof of Theorem 1 is finished.□

Next we prove Theorem 2.

Proof. For any odd number $q > 3$ with $(q, 3) = 1$ , we have

$\begin{eqnarray} S(1, q) = \sum\limits_{a = 1}^{q-1}\left(\frac{a}{q}-\frac{1}{2}\right)^2 = \frac{q}{12}-\frac{1}{4}+ \frac{1}{6q}. \end{eqnarray}$

(3.7)

On the basis of (3.7), Lemma 1 and the Möbius inversion formula, we can obtain

$\begin{eqnarray} \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = - 1}|L(1, \chi)|^2 = \frac{\pi^2}{12}\cdot \frac{\phi^2(q)}{q^2}\left[q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-3\right] \end{eqnarray}$

(3.8)

and

$\begin{eqnarray} \mathop{\sum\limits_{\chi ~ \text{mod}~ 3q }}_{\chi(-1) = - 1} |L(1, \chi)|^2 = \frac{\pi^2}{27}\cdot \frac{\phi^2(q)}{q^2}\left[4q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-3\right]. \end{eqnarray}$

(3.9)

According to the formulae (1.1), (3.3), (3.5) and the Möbius inversion formula, we also calculate the

$\begin{eqnarray} \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = - 1}\chi(3)\cdot |L(1, \chi)|^2 = \frac{\pi^2}{36}\cdot \frac{\phi^2(q)}{q^2}\left[q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-9-2\left(\frac{q}{3}\right)\cdot \prod\limits_{p\mid q}\frac{p-\left(\frac{p}{3}\right)}{p-1}\right]. \end{eqnarray}$

(3.10)

On the other hand, let $\lambda_3 = \left(\frac{*}{3}\right)$ denote the Legendre symbol modulo $3$ , and $\lambda_3^0$ denote the principal character modulo $3$ . Then note that

$\begin{align} \begin{aligned}\begin{split} \mathop{\sum\limits_{\chi ~ \text{mod}~ 3q}}_{\chi(-1) = -1} |L(1, \chi)|^2 & = \mathop{\sum\limits_{\chi ~ \text{mod}~ q}}_{\chi(-1) = -1} |L\left(1, \chi\lambda_3^0\right)|^2+\mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = 1} |L\left(1, \chi\lambda_3\right)|^2\\ & = \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = - 1} \left|1-\frac{\chi(3)}{3}\right|^2\cdot |L\left(1, \chi\right)|^2+\mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = 1} |L\left(1, \chi\lambda_3\right)|^2\\ & = \frac{10}{9}\cdot \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = - 1} |L(1, \chi)|^2- \frac{2}{3}\cdot \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = - 1} \chi(3)\cdot |L(1, \chi)|^2+\mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = 1} |L\left(1, \chi\lambda_3\right)|^2.\end{split}\end{aligned} \end{align}$

(3.11)

Then combining (3.8)–(3.11) we have

The proof of Theorem 2 is finished.□

Last, we will prove Theorem 3.

Proof. Combining with the formula (1.1), Lemma 1 and the Möbius inversion formula, we have

$\begin{eqnarray} \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = - 1}\chi(2)\cdot |L(1, \chi)|^2 = \frac{\pi^2}{24}\cdot \frac{\phi^2(q)}{q^2}\left[q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-6\right] \end{eqnarray}$

(3.12)

and

$\begin{eqnarray} \mathop{\sum\limits_{\chi ~ \text{mod}~ 3q }}_{\chi(-1) = - 1} \chi(2)\cdot |L(1, \chi)|^2 = \frac{\pi^2}{27}\cdot \frac{\phi^2(q)}{q^2}\left[2q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-3\right]. \end{eqnarray}$

(3.13)

For any positive integer $q$ with $(q, 6) = 1$ , note that the identity

$\begin{eqnarray*} S(q, 6) = \frac{5}{18}\cdot \left(\frac{q}{3}\right). \end{eqnarray*}$

We can calculate

$\begin{eqnarray} \mathop{\sum\limits_{\chi ~ \text{mod}~ q }}_{\chi(-1) = - 1} \chi(6)\cdot |L(1, \chi)|^2 = \frac{\pi^2}{72}\cdot \frac{\phi^2(q)}{q^2}\left[q\cdot \prod\limits_{p\mid q}\left(1+\frac{1}{p}\right)-18-20\left(\frac{q}{3}\right)\cdot\prod\limits_{p\mid q}\frac{p-\left(\frac{p}{3}\right)}{p-1} \right]. \end{eqnarray}$

(3.14)

On the other hand, let $\chi_q^0$ denote the principal character modulo $q$ and $\psi_3 = \left(\frac{*}{3}\right)$ . Then $\psi_3(2) = -1$ , so we also obtain

$\begin{align} \begin{split} \mathop{\sum\limits_{\chi ~ \text{mod}~ 3q }}_{\chi(-1) = - 1}\chi(2)\cdot |L(1, \chi)|^2 = & \mathop{\sum\limits_{\psi ~ \text{mod}~ 3}}_{\psi(-1) = 1}\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = -1}\lambda(2)\psi(2)\cdot \left|L\left(1, \lambda\psi\right)\right|^2\\ &+ \mathop{\sum\limits_{\psi ~ \text{mod}~ 3}}_{\psi(-1) = -1}\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = 1}\chi(2)\psi(2)\cdot \left|L\left(1, \lambda\psi\right)\right|^2\\ = &\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = -1}\chi(2)\cdot \left|L\left(1, \lambda\chi_3^0\right)\right|^2-\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = 1}\lambda(2)\left|L\left(1, \lambda\psi_3\right)\right|^2\\ = &\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = -1}\lambda(2)\cdot \left|1-\frac{\lambda(3)}{3}\right|^2\cdot \left|L\left(1, \lambda\right)\right|^2- \mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = 1}\lambda(2)\cdot \left|L\left(1, \lambda\psi_3\right)\right|^2\\ = &\frac{10}{9}\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = -1}\lambda(2)\cdot |L(1, \lambda)|^2-\frac{1}{3}\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = -1}\lambda(6)\cdot|L(1, \lambda)|^2\\ &-\frac{1}{3}\mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = -1}\lambda(2)\overline{\lambda}(3)\cdot|L(1, \lambda)|^2- \mathop{\sum\limits_{\lambda ~ \text{mod}~ q}}_{\lambda(-1) = 1}\lambda(2)\left|L\left(1, \lambda\psi_3\right)\right|^2.\end{split} \end{align}$

(3.15)

Now, combining Theorem 1 and (3.12)–(3.15), we have the identity

This completes the proof of Theorem 3.□

4. Conclusions

Our main purpose of this paper is to give a new reciprocity theorem for Dedekind sums (see Lemma 3). As an application of this result, we give a new calculating formula for one kind mean square value of Dirichlet $L$ -fuctions with the weight of the character sums. That is, we proved that for any positive integer $q > 1$ and $(q, 6) = 1$ , one has the identity

where $\left(\frac{*}{3}\right)$ denote the Legendre symbol modulo $3$ .

In addition, according to the reciprocity formula (1.1) and the Lemma 1, we may immediately deduce that for any two distinct odd primes $p$ and $q$ , one has the identity

$\begin{eqnarray} \frac{q}{q-1}\mathop{\sum\limits_{\chi ~ \text{mod}~ q}}_{\chi(-1) = -1} \chi(p)\cdot |L(1, \chi)|^2+ \frac{p}{p-1}\mathop{\sum\limits_{\chi ~ \text{mod}~ p}}_{\chi(-1) = -1} \chi(q)\cdot |L(1, \chi)|^2 = \frac{\pi^2}{12}\cdot \frac{p^2+q^2-3pq+1}{pq}. \end{eqnarray}$

(4.1)

Whether there exists a direct proof of (4.1) (using only the properties of Dirichlet $L$ -function, the reciprocity formula (1.1) cannot be used) is an open problem. It is believed that the methods used in this paper will contribute to further research in relevant fields.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the editor and referees for their very helpful and detailed comments.

This work is supported by the Natural Science Basic Research Plan in Shaanxi Province of China (2023-JC-QN-0050), Natural Science Foundation of China (12126357) and Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 22JK0424).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

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AIMS Mathematics

A new reciprocity formula of Dedekind sums and its applications

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Abstract

1. Introduction

2. Several lemmas

3. Proofs of the theorems

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

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A new reciprocity formula of Dedekind sums and its applications

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1. Introduction

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