A deep learning approach of financial distress recognition combining text

Jiawang Li; Chongren Wang; Jiawang Li; Chongren Wang

doi:10.3934/era.2023240

Electronic Research Archive

2023, Volume 31, Issue 8: 4683-4707. doi: 10.3934/era.2023240

Previous Article Next Article

Research article

A deep learning approach of financial distress recognition combining text

Jiawang Li ^,,
Chongren Wang ^,

School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan 250014, Shandong, China

Received: 05 April 2023 Revised: 04 June 2023 Accepted: 15 June 2023 Published: 03 July 2023

The financial distress of listed companies not only harms the interests of internal managers and employees but also brings considerable risks to external investors and other stakeholders. Therefore, it is crucial to construct an efficient financial distress prediction model. However, most existing studies use financial indicators or text features without contextual information to predict financial distress and fail to extract critical details disclosed in Chinese long texts for research. This research introduces an attention mechanism into the deep learning text classification model to deal with the classification of Chinese long text sequences. We combine the financial data and management discussion and analysis Chinese text data in the annual reports of 1642 listed companies in China from 2017 to 2020 in the model and compare the effects of the data on different models. The empirical results show that the performance of deep learning models in financial distress prediction overcomes traditional machine learning models. The addition of the attention mechanism improved the effectiveness of the deep learning model in financial distress prediction. Among the models constructed in this study, the Bi-LSTM+Attention model achieves the best performance in financial distress prediction.

Keywords:

Citation: Jiawang Li, Chongren Wang. A deep learning approach of financial distress recognition combining text[J]. Electronic Research Archive, 2023, 31(8): 4683-4707. doi: 10.3934/era.2023240

Related Papers:

[1]	Hongyan Dui, Yong Yang, Xiao Wang . Reliability analysis and recovery measure of an urban water network. Electronic Research Archive, 2023, 31(11): 6725-6745. doi: 10.3934/era.2023339
[2]	Xu Zhan, Yang Yong, Wang Xiao . Phased mission reliability analysis of unmanned ship systems. Electronic Research Archive, 2023, 31(10): 6425-6444. doi: 10.3934/era.2023325
[3]	Majed Alowaidi, Sunil Kumar Sharma, Abdullah AlEnizi, Shivam Bhardwaj . Integrating artificial intelligence in cyber security for cyber-physical systems. Electronic Research Archive, 2023, 31(4): 1876-1896. doi: 10.3934/era.2023097
[4]	Xiaoyang Xie, Shanghua Wen, Minglong Li, Yong Yang, Songru Zhang, Zhiwei Chen, Xiaoke Zhang, Hongyan Dui . Resilience evaluation and optimization for an air-ground cooperative network. Electronic Research Archive, 2024, 32(5): 3316-3333. doi: 10.3934/era.2024153
[5]	Shuang Yao, Dawei Zhang . A blockchain-based privacy-preserving transaction scheme with public verification and reliable audit. Electronic Research Archive, 2023, 31(2): 729-753. doi: 10.3934/era.2023036
[6]	Shanpu Gao, Yubo Li, Anping Wu, Hao Jiang, Feng Liu, Xinlong Feng . An intelligent optimization method for accelerating physical quantity reconstruction in computational fluid dynamics. Electronic Research Archive, 2025, 33(5): 2881-2924. doi: 10.3934/era.2025127
[7]	Wanxun Jia, Ling Li, Haoyan Zhang, Gengxiang Wang, Yang Liu . A novel nonlinear viscous contact model with a Newtonian fluid-filled dashpot applied for impact behavior in particle systems. Electronic Research Archive, 2025, 33(5): 3135-3157. doi: 10.3934/era.2025137
[8]	Sida Lin, Jinlong Yuan, Zichao Liu, Tao Zhou, An Li, Chuanye Gu, Kuikui Gao, Jun Xie . Distributionally robust parameter estimation for nonlinear fed-batch switched time-delay system with moment constraints of uncertain measured output data. Electronic Research Archive, 2024, 32(10): 5889-5913. doi: 10.3934/era.2024272
[9]	Yunying Huang, Wenlin Gui, Yixin Jiang, Fengyi Zhu . Types of systemic risk and macroeconomic forecast: Evidence from China. Electronic Research Archive, 2022, 30(12): 4469-4492. doi: 10.3934/era.2022227
[10]	Yazhou Chen, Dehua Wang, Rongfang Zhang . On mathematical analysis of complex fluids in active hydrodynamics. Electronic Research Archive, 2021, 29(6): 3817-3832. doi: 10.3934/era.2021063

Abstract

1. Introduction

Let $q$ be a power of odd prime. Several researchers have looked into a variety of properties about the primitive roots modulo $q$ . Let $g_1, g_2$ represent two primitive roots modulo $q$ , $a$ , $b$ and $c$ represent arbitrary non-zero elements in $\mathbb{F}_q$ . Is there some $q_0$ such that for all $q > q_0$ , there is always one representation

$\begin{eqnarray} a = bg_1+cg_2\ ? \end{eqnarray}$

(1.1)

For $b = 1$ and $c = -1$ , Vegh ^[1] considered a specific form of Eq (1.1), which is known as Vegh's Conjecture, (see [,§ F9] for further details). Cohen ^[3] demonstrated Vegh's Conjecture for all $q > 7$ .

For $b = 1$ and $c = 1$ , Golomb ^[4] proposed another specific form of Eq (1.1). This was proved by Sun ^[5] for $q > 2^{60}\approx 1.15\times 10^{18}$ .

Moreover, Cohen et al. ^[6] studied linear sums of primitive roots and their inverses in finite fields $\mathbb{F}_q$ and showed that if $q > 13$ , then for arbitrary non-zero $a, b \in \mathbb{F}_q$ , there is a pair of primitive elements ( $g_1$ , $g_2$ ) of $\mathbb{F}_q$ such that both $ag_1+bg_2$ and $ag^{-1}_1+bg^{-1}_2$ are primitive.

Let $p$ be an odd prime. Carlitz ^[7] relied on some results of Davenport and obtained for any $k-1$ fixed integers $c_1, c_2, \ldots, c_{k-1}$ with $c_i \geq 1 (i = 1, 2, \ldots, k-1)$ . Let $g, g_1, \ldots, g_{k-1}$ be primitive roots modulo $p$ and $N_k$ denote the number of $g\bmod p$ such that $g_1-g = c_1, \ldots, g_{k-1}-g = c_{k-1}$ . Then

$\begin{eqnarray*} N_k \sim \frac{\phi^{k}(p-1)}{p^{k-1}}\ \hfill (p\rightarrow \infty). \end{eqnarray*}$

More results of the primitive roots distribution can be found in ^[8,9,10,11].

Lehmer [,§ F12] proposed the definition of $Lehmer~~ number$ , according to which $a$ is a $Lehmer ~~number$ if and only if $a$ and $\bar{a}$ have opposite parity, i.e., $(2, a+\bar{a}) = 1$ , where $\bar{a}$ is the multiplicative inverse of $a$ modulo $p$ . It is simple to demonstrate that there are no Lehmer numbers modulo $p$ when $p = 3$ or $7$ . Zhang ^[12] established that if $M_p$ denotes the number of Lehmer numbers modulo $p$ , then

$\begin{eqnarray*} M_p = \frac{p-1}{2}+\mathrm{O}\left(p^{\frac{1}{2}}\ln^2p\right). \end{eqnarray*}$

A Lehmer number that is also a primitive root modulo $p$ will be called a $Lehmer~~ primitive~~ root$ or an $LPR$ . The inverse of an $LPR$ is also an $LPR$ . We assume that $p > 3$ because there is no Lehmer number modulo $3$ . Wang and Wang ^[13] investigated the distribution of $LPRs$ involving Golomb's conjecture. Let $G_p$ denote the number of Golomb pairs $(a, b)$ (i.e., $a+b\equiv 1(\bmod\; p)$ ) are $LPRs$ . They showed

$\begin{eqnarray*} G_p = \frac{1}{4}\frac{\phi^2(p-1)}{p-1}+O\left(\frac{\phi^2(p-1)}{p^{\frac{5}{4}}}\cdot 4^{\omega(p-1)}\cdot \ln^2p\right). \end{eqnarray*}$

Let $N_p$ denote the number of $LPRs$ modulo $p$ . For odd integers $m\geq 3$ , define the positive number $T_m$ by

$\begin{eqnarray*} T_m = \frac{2}{m\ln m}\sum\limits_{j = 1}^{(m-1)/2}\tan\left(\frac{\pi j}{m}\right). \end{eqnarray*}$

Cohen and Trudgian ^[14] improved the result of Wang and Wang ^[13] and showed

$\begin{eqnarray*} \left|N_p-\frac{\phi(p-1)}{2}\right| < T_p^2\frac{\phi(p-1)}{p-1}2^{\omega(p-1)}p^{\frac{1}{2}}\ln^2p \end{eqnarray*}$

and

$\begin{eqnarray*} \left|G_p-\frac{\phi^2(p-1)}{4(p-1)^2}(p-2)\right| < \frac{\phi^2(p-1)}{4(p-1)^2}T_p^2[2^{2\omega(p-1)}(9\ln^2p+1)-1]p^{\frac{1}{2}}, \end{eqnarray*}$

where $\frac{2}{\pi}\left(1+\frac{0.548}{\ln p}\right) < T_p < \frac{2}{\pi}\left(1+\frac{1.549}{\ln p}\right)$ .

Specifically, they obtained that for an odd prime $p(\neq 3, 7)$ , there exists an $LPR$ modulo $p$ .

Inspired by the results of Cohen and Trudgian ^[14] and Wang and Wang ^[13], we mainly studied the distribution of $LPRs$ modulo $p$ related to the Golomb's conjecture in two aspects. On the one hand, we extend Eq (1.1) to the case involving $k > 1$ variables. Let $\mathcal{R}$ be set of $LPRs$ modulo $p$ that is a subset of ${\mathbb{F}}_p$ . $a_1, a_2, \ldots, a_k, c$ are non-zero elements in ${\mathbb{F}}_p$ and $N(\mathcal{R}, p)$ denotes the number of solutions of the equation

$\begin{eqnarray*} a_1g_1+a_2g_2+\cdots+a_kg_k = c, \ g_1, g_2, \ldots, g_k \in \mathcal{R}. \end{eqnarray*}$

We consider the distribution properties of $N(\mathcal{R}, p)$ , and obtain the following:

Theorem 1. Let $p > 3$ be an odd prime. Then we have

$\begin{eqnarray*} N(\mathcal{R}, p) = \frac{\phi^k(p-1)}{2^kp}+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^\frac{3}{2}}2^{k\omega(p-1)}\ln^{2k}p\right), \end{eqnarray*}$

where the symbol $\mathrm{O}$ is dependent on $k$ .

When $k = 2$ , we can obtain the number of the Golomb pairs that are $LPRs$ .

On the other hand, we consider the distribution of $k$ consecutive $LPRs$ and generalize it to a more general form.

Let $f(x)\in \mathbb{F}_p[x]$ . Define

$\begin{eqnarray*} M(f(x), \mathcal{R}, p) = \#\{x: 1\leq x\leq p-1, f(x+c_1), f(x+c_2), \cdots, f(x+c_k) \in \mathcal{R} \}. \end{eqnarray*}$

Then we have:

Theorem 2. Let $f(x)\in \mathbb{F}_p[x]$ with degree $l\geq 1$ . $c_1, c_2, \ldots, c_k$ are distinct elements in $\mathbb{F}_p$ . Suppose that one of the following conditions holds:

$\rm (i)$ $f(x)$ is irreducible,

$\rm (ii)$ $f(x)$ has no multiple zero in $\bar{\mathbb{F}}_p$ and $k = 2$ ,

$\rm (iii)$ $f(x)$ has no multiple zero in $\bar{\mathbb{F}}_p$ and $(4k)^{l} < p$ .

Then we have

$\begin{eqnarray*} &&M(f(x), \mathcal{R}, p) = \frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^{k-1}}+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^{k-\frac{1}{2}}}2^{k\omega(p-1)}\ln^{2k}p\right), \end{eqnarray*}$

where the symbol $\mathrm{O}$ is dependent on $k$ and $l$ .

Take $f(x) = x$ , $c_k = 0$ in Theorem 2. Then we can get the number of $k$ consecutive primitive roots $x, x+c_1, \ldots, x+c_{k-1}$ are Lehmer numbers, which is:

Corollary 1. Let $p$ be an odd prime. Then for any $1\leq\ x\leq(p-1)$ that is an LPR modulo $p$ , we have

$\begin{eqnarray*} &&M(x, \mathcal{R}, p) = \frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^{k-1}}+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^{k-\frac{1}{2}}}2^{k\omega(p-1)}\ln^{2k}p\right), \end{eqnarray*}$

where the symbol $\mathrm{O}$ is dependent on $k$ .

When $k = 1, 2$ , we can easily deduce the Theorem 1 and Theorem 6 in Cohen and Trudgian ^[14], respectively.

Notation: Throughout this paper, $\mathbb{F}_q$ denotes a finite field of characteristic $p$ , $\bar{\mathbb{F}}_q$ denotes the algebraic closure of $\mathbb{F}_q$ , $\phi(n)$ is reserved for the Euler function, $\mu(n)$ is the M $\ddot{o}$ bius function. We use $\omega(n)$ to denote the number of all distinct prime divisors of $n$ . Write $\sum_{\chi_d}$ to denote a sum over all $\phi(d)$ multiplicative characters $\chi_d$ of order $d$ over ${\mathbb{F}}_p$ , and denote by $\mathop{\sum_{n = 1}^{p}}^{'}$ the summation of $1\leq n\leq p$ with $(n, p) = 1$ . $\tau(\chi)$ is the classical Gauss sums associated with character $\chi$ mudulo $p$ . $f\ll g$ means $|f|\leq cg$ with some positive constant $c$ , $f = \mathrm{O}(g)$ means $f\ll g$ .

2. Some lemmas

To complete the proof of the theorems, we need following several lemmas. The proofs of these lemmas require some basic knowledge of analytic number theory, which can be found in ^[15].

Lemma 1. Let $p$ be an odd prime. Then for any integer $a$ coprime to $p$ (i.e., $(a, p) = 1$ ), we have the identity

$\frac{\phi(p-1)}{p-1}\sum\limits_{d\mid p-1}\frac{\mu(d)}{\phi(d)}\sum\limits_{\chi_d}\chi_d(a) = \left\{ \begin{array}{ll} 1, &\ { if~~ a ~~ is ~~ a~~ primitive ~~ root ~~ \text{mod}~~ p;}\\ 0, &\ {if~~ a ~~ is~~ not~~ a~~ primitive~~ root~~ \text{mod}~~ p.} \end{array}\right.$

Proof. See Proposition 2.2 of Narkiewicz ^[16]. □

Lemma 2. Let $p$ be an odd prime, $\chi$ be a nonprincipal multiplicative character modulo $p$ of order $d$ . Suppose $g(x)\in \mathbb{F}_p[x]$ has precisely $m$ distinct ones among its zeros, and suppose that $g(x)$ is not the constant multiple of a $d$ -th power over $\mathbb{F}_q$ . Then

$\left|\sum\limits_{x\in \mathbb{F}_p}\chi\left(g(x)\right)\right|\leq \left(m-1\right)\cdot p^{\frac{1}{2}}.$

Proof. See Theorem 2C in Chapter 2 of Schmidt ^[17]. □

Lemma 3. Let $\mathbb{F}_q$ be a finite field of characteristic $p$ , $\psi$ be a nontrivial additive character and $\chi$ be a nonprincipal multiplicative character on $\mathbb{F}_q$ of order $d$ . For two rational functions $f(x), g(x)\in\mathbb{F}_q[x]$ , define $K(\psi, f;\chi, g) = \sum_{x\in\mathbb{F}_q\backslash S}\chi(g(x))\psi(f(x))$ , where $S$ denotes the set of poles of $f(x)$ and $g(x)$ . Suppose the following conditions hold:

$\rm (i)$ $g(x)$ is not the constant multiple of a $d$ -th power over $\mathbb{F}_q$ .

$\rm (ii)$ $f(x)$ is not of the form $(h(x))^p-h(x)$ with a rational function $h(x)$ over $\mathbb{F}_q$ .

Then we have

$|K(\psi, f;\chi, g)|\leq(\deg(f)+m-1)\sqrt{q},$

where $m$ is the number of distinct roots and (noninfinite) poles of $g(x)$ in $\mathbb{F}_q$ .

Proof. See Theorem 2G in Chapter 2 of Schmidt ^[17]. □

Lemma 4. Let $p$ be an odd prime. Let $c_1, \cdots, c_k$ be distinct elements in $\mathbb{F}_p$ . Assume that $f(x)\in\mathbb{F}_{p}[x]$ with $\deg(f) = l$ . Define the polynomial

$h(x) = f(x+c_1)\cdots f(x+c_k).$

Suppose one of the following conditions holds:

$\rm (i)$ $f(x)$ is irreducible,

$\rm (ii)$ $f(x)$ has no multiple zero in $\bar{\mathbb{F}}_p$ and $k = 2$ ,

$\rm (iii)$ $f(x)$ has no multiple zero in $\bar{\mathbb{F}}_p$ and $(4k)^{l} < p$ .

Then $h(x)$ has at least one simple root in $\bar{\mathbb{F}}_p$ .

Proof. Suppose that $f(x)$ is irreducible. Then $f(x+c_1), \cdots, f(x+c_k)$ are distinct irreducible polynomials, and $h(x)$ has at least $k$ simple roots in $\bar{\mathbb{F}}_p$ . The cases of $\rm (ii)$ and $\rm (iii)$ can be proved by Theorem 2 and Lemma 2 of ^[18], for $k = 2$ or $(4k)^{l} < p$ , $(l, k, p)$ is "admissible triple, " then $f(x+c_1)\cdots f(x+c_k)$ has at least one simple root. □

Lemma 5. Let $p$ be an odd prime, $m_1, \ldots, m_k, n_1, \ldots, n_k$ be integers with $(m_1\cdots m_kn_1\cdots n_k, p) = 1$ , and polynomials $g(x), f_1(x), \ldots, f_k(x)\in\mathbb{F}_p[x]$ . Let $\chi$ be a Dirichlet character modulo $p$ of order $d$ . Define

$\begin{eqnarray*} &&K(\chi, g, f_1, \cdots, f_k;p)\\ && = \mathop{\sum\limits_{x = 1}^{p}}_{(f_1(x)\cdots f_k(x), p) = 1}\chi(g(x)) e\left(\frac{m_1f_1(x)+\cdots+m_kf_k(x)+n_1\overline{f_1(x)}+\cdots+n_k\overline{f_k(x)}}{p}\right). \end{eqnarray*}$

Suppose the following conditions hold:

$\rm (i)$ $g(x)$ can not be the constant multiple of a $d$ -th power over $\mathbb{F}_p$ .

$\rm (ii)$ $F(x) = f_1(x)\cdots f_k(x)$ has at least one simple root in $\bar{\mathbb{F}}_p$ .

Then we have

$\begin{eqnarray*} |K(\chi, g, f_1, \cdots, f_k;p)| \leq\left(\max(\deg(f_1), \cdots, \deg(f_k))+l\right)\sqrt{p}, \end{eqnarray*}$

where $e(x) = e^{2\pi ix}$ and $l$ is the number of distinct roots of $g(x)$ in $\bar{\mathbb{F}}_p$ .

Proof. It is clear that

$\begin{eqnarray*} &&m_1f_1(x)+\cdots+m_kf_k(x)+n_1\overline{f_1(x)}+\cdots+n_k\overline{f_k(x)}\\ && = \frac{F(x)(m_1f_1(x)+\cdots+m_kf_k(x))+n_1\frac{F(x)}{f_1(x)}+\cdots +n_k\frac{F(x)}{f_k(x)}}{F(x)} : = \frac{G(x)}{F(x)}. \end{eqnarray*}$

Condition $\rm (i)$ is the same as Lemma 3. So our goal is to prove the rational function $G(x)/F(x)$ satisfies condition $\rm (ii)$ in Lemma 3 if $F(x)$ has a simple root in $\bar{\mathbb{F}}_p$ . Assume that there are polynomials $K(x), L(x)\in\mathbb{F}_p[x]$ with $(K(x), L(x)) = 1$ such that

$\frac{G(x)}{F(x)} = \left(\frac{K(x)}{L(x)}\right)^p-\left(\frac{K(x)}{L(x)}\right).$

Then we have

$\begin{eqnarray} G(x)L(x)^p = \left(K(x)^p-K(x)L(x)^{p-1}\right)F(x). \end{eqnarray}$

(2.1)

Since $F(x) = f_1(x)\cdots f_k(x)$ has at least one simple root in $\bar{\mathbb{F}}_p$ , then there exists an irreducible polynomial $w(x)\in\mathbb{F}_p[x]$ such that $w(x)\mid F(x)$ and $w(x)^2\nmid F(x)$ . Assume that $w(x)\mid f_1(x)$ , then we have

$w(x)\nmid\frac{F(x)}{f_1(x)}, \ w(x)\mid\frac{F(x)}{f_i(x)}(i = 2, \cdots, k).$

Hence, from Eq (2.1)

$w(x)\nmid G(x)\Longrightarrow w(x)\mid L(x)^p\Longrightarrow w(x)\mid L(x)$

$w(x)^2\mid L(x)^{p-1}\Longrightarrow w(x)^2\mid K(x)^pF(x)\Longrightarrow w(x)\mid K(x),$

which contradicts to $(K(x), L(x)) = 1$ . Therefore, from Lemma 3 we get

$\begin{eqnarray*} |K(\chi, g, f_1, \cdots, f_k;p)| \leq\left(\max(\deg(f_1), \cdots, \deg(f_k))+l\right)\sqrt{p}, \end{eqnarray*}$

where $l$ is the number of distinct roots of $g(x)$ in $\bar{\mathbb{F}}_p$ . □

Lemma 6. Let $\chi$ be a primitive character modulo $p$ , $\chi_{d_i}$ be character modulo $p$ of order $d_i$ . There exist some $1\leq s_i\leq d_i$ with $(s_i, d_i) = 1$ , $i = 1, 2, \ldots, k$ . Then we have

Proof. From the definition of the Dirichlet character modulo $p$ , we can get

$\begin{eqnarray*} &&\sum\limits_{\chi_{d_1}}\cdots \sum\limits_{\chi_{d_k}}\chi_{d_1}(f(x+c_1))\cdots \chi_{d_k}(f(x+c_k))\\ & = &\mathop{{\sum\limits_{s_1 = 1}^{d_1}}\ '}\cdots \mathop{{\sum\limits_{s_k = 1}^{d_k}}\ '} e\left(\frac{s_1\cdot \text{ind}(f(x+c_1))}{d_1}\right)\cdots e\left(\frac{s_k\cdot \text{ind}(f(x+c_k))}{d_k}\right)\\ & = &\mathop{{\sum\limits_{s_1 = 1}^{d_1}}\ '}\cdots \mathop{{\sum\limits_{s_k = 1}^{d_k}}\ '}e\left(\frac{\frac{s_1(p-1)}{d_1}\cdot \text{ind}(f(x+c_1))+\cdots+\frac{s_k(p-1)}{d_k}\cdot \text{ind}(f(x+c_k))}{p-1}\right)\\ & = &\mathop{{\sum\limits_{s_1 = 1}^{d_1}}\ '}\cdots \mathop{{\sum\limits_{s_k = 1}^{d_k}}\ '}e\left(\frac{ \text{ind}(f(x+c_1))^{\frac{s_1(p-1)}{d_1}}+\cdots+ \text{ind}(f(x+c_k))^{\frac{s_k(p-1)}{d_k}}}{p-1}\right)\\ & = &\mathop{{\sum\limits_{s_1 = 1}^{d_1}}\ '}\cdots \mathop{{\sum\limits_{s_k = 1}^{d_k}}\ '}e\left(\frac{ \text{ind}\left((f(x+c_1))^{\frac{s_1(p-1)}{d_1}}\cdots(f(x+c_k))^{\frac{s_k(p-1)}{d_k}}\right)}{p-1}\right)\\ & = &\mathop{{\sum\limits_{s_1 = 1}^{d_1}}\ '}\cdots \mathop{{\sum\limits_{s_k = 1}^{d_k}}\ '}\chi\left(f(x+c_1))^{\frac{s_1(p-1)}{d_1}}\cdots(f(x+c_k))^{\frac{s_k(p-1)}{d_k}}\right), \end{eqnarray*}$

where $\text{ind}(a)$ denotes an index of $a$ with base $g$ of modulo $p$ , and $g$ is a positive primitive root of modulo $p$ . □

3. Proofs of the theorems

Firstly, we prove the Theorem 1. Let $p$ be an odd prime, $k$ be any fixed positive integer. Then for any $k$ different integers $a_1, \ a_2, \ldots, a_k \in {\mathbb{F}}_p$ , from Lemma 1 and the definition of Lehmer number we have

$\begin{eqnarray} &&N(\mathcal{R}, p) = \frac{1}{p}\sum\limits_{b = 0}^{p-1}\mathop{\sum\limits_{g_1 = 1}^{p-1}\sum\limits_{g_2 = 1}^{p-1}\cdots \sum\limits_{g_k = 1}^{p-1}}_{g_1, g_2, \ldots, g_k \in \mathcal{R}}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\\ & = &\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\sum\limits_{g_i = 1}^{p-1} \chi_{d_i}(g_i)\left(1-(-1)^{g_i+\overline{g_i}}\right)\right)\\ &&\cdot \sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\\ & = &\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\sum\limits_{g_i = 1}^{p-1} \chi_{d_i}(g_i)\right)\sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\\ &&+\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\sum\limits_{g_i = 1}^{p-1} \chi_{d_i}(g_i)\right)\sum\limits_{t = 1}^{k}(-1)^{t}\mathop{\sum\limits_{i_1 = 1}^{k}\sum\limits_{i_2 = 1}^{k}\cdots \sum\limits_{i_t = 1}^{k}}_{i_1 < i_2 < \cdots < i_t}l_{i_1}l_{i_2}\cdots l_{i_t}\\ &&\cdot \sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\\ & = &A_1+A_2, \end{eqnarray}$

(3.1)

where $l_i = (-1)^{g_i+\overline{g_i}}, i = 1, 2, \cdots, k$ .

$\begin{eqnarray} &&A_1 = \frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\sum\limits_{g_i = 1}^{p-1} \chi_{d_i}(g_i)\right)\sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\\ & = &\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\left[\sum\limits_{g_1 = 1}^{p-1}\cdots \sum\limits_{g_k = 1}^{p-1}\sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right.\\ &&+\left.\mathop{\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}}_{d_1\cdots d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{g_1 = 1}^{p-1}\cdots \sum\limits_{g_k = 1}^{p-1} \chi_{d_1}(g_1)\cdots \chi_{d_k}(g_k)\right.\\ &&\cdot\left.\sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right]\\ & = &\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\left[(p-1)^k+(-1)^{k+1}+\mathop{\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}}_{d_1\cdots d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\right.\\ &&\cdot\left.\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{g_1 = 1}^{p-1}\cdots \sum\limits_{g_k = 1}^{p-1} \chi_{d_1}(g_1)\cdots \chi_{d_k}(g_k)\sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right]. \end{eqnarray}$

(3.2)

From Eq (3.2), let

$\begin{eqnarray*} A_{11}& = &\mathop{\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}}_{d_1\cdots d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{g_1 = 1}^{p-1}\cdots \sum\limits_{g_k = 1}^{p-1} \chi_{d_1}(g_1)\cdots \chi_{d_k}(g_k)\\ &&\cdot\sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+a_2g_2+\cdots+a_kg_k-c)}{p}\right)\\ & = &\mathop{\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}}_{d_1\cdots d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{g_1 = 1}^{p-1}\cdots \sum\limits_{g_k = 1}^{p-1} \chi_{d_1}(g_1)\cdots \chi_{d_k}(g_k)\\ &&+\mathop{\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}}_{d_1\cdots d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{b = 1}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\chi_{d_1}(g_1)e\left(\frac{ba_1g_1}{p}\right)\\ &&\cdots \sum\limits_{g_k = 1}^{p-1}\chi_{d_k}(g_k)e\left(\frac{ba_kg_k}{p}\right)e\left(\frac{-bc}{p}\right)\\ & = &\mathop{\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}}_{d_1\cdots d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{b = 1}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\chi_{d_1}(g_1)e\left(\frac{ba_1g_1}{p}\right)\\ &&\cdots \sum\limits_{g_k = 1}^{p-1}\chi_{d_k}(g_k)e\left(\frac{ba_kg_k}{p}\right)e\left(\frac{-bc}{p}\right). \end{eqnarray*}$

Using the properties of Gauss sums we can get

$\begin{eqnarray*} \left|A_{11}\right|& = &\left|\mathop{\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}}_{d_1\cdots d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{b = 1}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\chi_{d_1}(g_1)e\left(\frac{ba_1g_1}{p}\right)\right.\\ &&\left.\cdots \sum\limits_{g_k = 1}^{p-1}\chi_{d_k}(g_k)e\left(\frac{ba_kg_k}{p}\right)e\left(\frac{-bc}{p}\right)\right|\\ & = &\left|\mathop{\sum\limits_{d_1\mid p-1}}_{d_1 > 1}\cdots \mathop{\sum\limits_{d_k\mid p-1}}_{d_k > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_k\right)}{\phi\left(d_k\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\sum\limits_{b = 1}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\chi_{d_1}(g_1)e\left(\frac{ba_1g_1}{p}\right)\right.\\ &&\cdots \sum\limits_{g_k = 1}^{p-1}\chi_{d_k}(g_k)e\left(\frac{ba_kg_k}{p}\right)e\left(\frac{-bc}{p}\right)\\ &&+\mathop{\sum\limits_{d_1\mid p-1}}_{d_1 > 1}\cdots \mathop{\sum\limits_{d_{k-1}\mid p-1}}_{d_{k-1} > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\cdots\frac{\mu\left(d_{k-1}\right)}{\phi\left(d_{k-1}\right)}\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_{k-1}}}\sum\limits_{b = 1}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\chi_{d_1}(g_1)e\left(\frac{ba_1g_1}{p}\right)\\ &&\cdots \sum\limits_{g_{k-1} = 1}^{p-1}\chi_{d_{k-1}}(g_{k-1})e\left(\frac{ba_{k-1}g_{k-1}}{p}\right)\sum\limits_{g_k = 1}^{p-1}e\left(\frac{ba_kg_k}{p}\right)e\left(\frac{-bc}{p}\right)\\ &&+\cdots+\mathop{\sum\limits_{d_1\mid p-1}}_{d_1 > 1}\frac{\mu\left(d_1\right)}{\phi\left(d_1\right)}\sum\limits_{\chi_{d_1}}\sum\limits_{b = 1}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\chi_{d_1}(g_1)e\left(\frac{ba_1g_1}{p}\right)\sum\limits_{g_2 = 1}^{p-1}e\left(\frac{ba_2g_2}{p}\right)\\ &&\left.\cdots \sum\limits_{g_k = 1}^{p-1}e\left(\frac{ba_kg_k}{p}\right)e\left(\frac{-bc}{p}\right)\right|\\ &\ll&2^{k\omega(p-1)}p^{\frac{k+1}{2}}, \end{eqnarray*}$

where we have used the fact that $\sum_{d|n}|\mu(d)| = 2^{\omega(n)}$ .

Hence, Eq (3.2) and the above formulae yield that

$\begin{eqnarray} A_1 = \frac{\phi^k(p-1)}{2^kp}+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^{\frac{k+1}{2}}}2^{k\omega(p-1)}\right). \end{eqnarray}$

(3.3)

Then we compute $A_2$ in Eq (3.1). For simplicity, let

$U_m(u) = \sum\limits_{u = 1}^{p-1}(-1)^ue\left(\frac{-mu}{p}\right),$

noting that

$\sum\limits_{u = 1}^{p-1}(-1)^ue\left(\frac{-mu}{p}\right) = \frac{1-e(\frac{m}{p})}{1+e(\frac{m}{p})} = \frac{i\sin (\pi m/p)}{\cos(\pi m/p)},$

$\sum\limits_{m = 1}^{p-1}\left|\frac{\sin (\pi m/p)}{\cos(\pi m/p)}\right| = T_pp\ln p.$

Hence,

$\begin{eqnarray} \left|\sum\limits_{m = 1}^{p-1}U_m(u)\right|\leq\sum\limits_{m = 1}^{p-1}\left|\sum\limits_{u = 1}^{p-1}(-1)^ue\left(\frac{-mu}{p}\right)\right| = T_pp\ln p. \end{eqnarray}$

(3.4)

Noting that, if $m = 0$ , then $\sum_{u = 1}^{p-1}(-1)^ue\left(\frac{-mu}{p}\right) = \sum_{u = 1}^{p-1}(-1)^u = 0$ , since $p$ is odd. Hence,

$\begin{eqnarray} &&l_i = (-1)^{g_i+\overline{g_i}}\\ & = &\frac{1}{p}\sum\limits_{m_i = 0}^{p-1}\sum\limits_{u_i = 1}^{p-1}(-1)^{u_i}e\left(\frac{m_i(g_i-u_i)}{p}\right)\cdot \frac{1}{p}\sum\limits_{n_i = 0}^{p-1}\sum\limits_{v_i = 1}^{p-1}(-1)^{v_i}e\left(\frac{n_i(\overline{g_i}-v_i)}{p}\right)\\ & = &\frac{1}{p^2}\sum\limits_{m_i, n_i = 0}^{p-1}e\left(\frac{m_ig_i+n_i\overline{g_i}}{p}\right)\sum\limits_{u_i = 1}^{p-1}(-1)^{u_i}e\left(\frac{-m_iu_i}{p}\right)\sum\limits_{v_i = 1}^{p-1}(-1)^{v_i}e\left(\frac{-n_iv_i}{p}\right)\\ & = &\frac{1}{p^2}\sum\limits_{m_i, n_i = 1}^{p-1}e\left(\frac{m_ig_i+n_i\overline{g_i}}{p}\right)U_{m_i}(u_i)U_{n_i}(v_i). \end{eqnarray}$

(3.5)

From the above discussion and Eq (3.1), we can obtain

$\begin{eqnarray} \left|A_2\right|& = &\left|\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\sum\limits_{g_i = 1}^{p-1} \chi_{d_i}(g_i)\right)\sum\limits_{t = 1}^{k}(-1)^{t}\mathop{\sum\limits_{i_1 = 1}^{k}\cdots \sum\limits_{i_t = 1}^{k}}_{i_1 < \cdots < i_t}l_{i_1}\cdots l_{i_t}\right.\\ &&\left.\cdot \sum\limits_{b = 0}^{p-1}e\left(\frac{b(a_1g_1+a_2g_2+\cdots+a_kg_k-c)}{p}\right)\right|\\ &\leq&\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\sum\limits_{t = 1}^{k}\binom{k}{t}T^{2t}_p\ln^{2t}p\sum\limits_{d_1\mid p-1}\cdots \sum\limits_{d_k\mid p-1}\frac{|\mu\left(d_1\right)|}{\phi\left(d_1\right)}\cdots\frac{|\mu\left(d_k\right)|}{\phi\left(d_k\right)}\\ &&\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}\chi_{d_1}(g_1)\cdots \chi_{d_k}(g_k)\right.\\ &&\left.\cdot e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_tg_t+n_t\overline{g_t}}{p}\right)e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\\ & = &\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\sum\limits_{t = 1}^{k}\binom{k}{t}T^{2t}_p\ln^{2t}p\left[\mathop{\sum\limits_{d_1\mid p-1}}_{d_1 > 1}\cdots \mathop{\sum\limits_{d_k\mid p-1}}_{d_k > 1}\frac{|\mu\left(d_1\right)|}{\phi\left(d_1\right)}\cdots\frac{|\mu\left(d_k\right)|}{\phi\left(d_k\right)}\right.\\ &&\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_k}}\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}\chi_{d_1}(g_1)\cdots \chi_{d_k}(g_k)\right.\\ &&\left.\cdot e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_tg_t+n_t\overline{g_t}}{p}\right)e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\\ &&+\mathop{\sum\limits_{d_1\mid p-1}}_{d_1 > 1}\cdots \mathop{\sum\limits_{d_{k-1}\mid p-1}}_{d_{k-1} > 1}\frac{|\mu\left(d_1\right)|}{\phi\left(d_1\right)}\cdots\frac{|\mu\left(d_{k-1}\right)|}{\phi\left(d_{k-1}\right)}\\ &&\sum\limits_{\chi_{d_1}}\cdots\sum\limits_{\chi_{d_{k-1}}}\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}\chi_{d_1}(g_1)\cdots \chi_{d_{k-1}}(g_{k-1})\right.\\ &&\left.\cdot e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_tg_t+n_t\overline{g_t}}{p}\right)e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\\ &&+\cdots+\mathop{\sum\limits_{d_1\mid p-1}}_{d_1 > 1}\frac{|\mu\left(d_1\right)|}{\phi\left(d_1\right)}\sum\limits_{\chi_{d_1}}\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}\chi_{d_1}(g_1)\right.\\ &&\left.\cdot e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_tg_t+n_t\overline{g_t}}{p}\right)e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\\ &&+\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_tg_t+n_t\overline{g_t}}{p}\right)\right.\\ &&\left.\left.\cdot e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\right]. \end{eqnarray}$

(3.6)

Summing the above formula for $t$ from 1 to $k$ , then the last term of Eq (3.6) is

$\begin{eqnarray*} &&\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\sum\limits_{t = 1}^{k}\binom{k}{t}T^{2t}_p\ln^{2t}p\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_tg_t+n_t\overline{g_t}}{p}\right)\right.\\ &&\left.\cdot e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\\ & = &\frac{1}{p}\frac{\phi^k(p-1)}{2^k(p-1)^k}\left[kT^2_p\ln^{2}p\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}e\left(\frac{m_1g_1+n_1\overline{g_1}}{p}\right)e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\right.\\ &&+\cdots+\binom{k}{k-1}T^{2(k-1)}_p\ln^{2(k-1)}p\cdot\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}\right.\\ &&\left.\cdot e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_{k-1}g_{k-1}+n_{k-1}\overline{g_{k-1}}}{p}\right)e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\\ &&+T^{2k}_p\ln^{2k}p\left|\sum\limits_{b = 0}^{p-1}\sum\limits_{g_1 = 1}^{p-1}\cdots\sum\limits_{g_k = 1}^{p-1}e\left(\frac{m_1g_1+n_1\overline{g_1}+\cdots+m_kg_k+n_k\overline{g_k}}{p}\right)\right.\\ &&\left.\left.\cdot e\left(\frac{b(a_1g_1+\cdots+a_kg_k-c)}{p}\right)\right|\right]\\ &\ll&\frac{\phi^k(p-1)}{p^{k+1}}\ln^{2k}p\left(p^{k-\frac{1}{2}}+\cdots+p^{\frac{k+1}{2}}\right)\\ &\ll&\frac{\phi^k(p-1)}{p^{k+1}}\ln^{2k}p\cdot p^{k-\frac{1}{2}} = \frac{\phi^k(p-1)}{p^{\frac{3}{2}}}\ln^{2k}p, \end{eqnarray*}$

here we have utilized $T^2_p < \frac{4}{\pi^2}\left(1+\frac{1.549}{\ln p}\right)^2 < 2.4$ and the results in Wang and Wang (see Lemma 2.2 of ^[13]) that

$\begin{eqnarray*} \left|\sum\limits_{a = 1}^{p-1}\chi_d(a)e\left(\frac{ma+n\overline{a}}{p}\right)\right|\ll p^{\frac{1}{2}}. \end{eqnarray*}$

Similarly, note that $\sum_{d|n}|\mu(d)| = 2^{\omega(n)}$ and we can get the estimate of the other terms of Eq (3.6). Then we have

$\begin{eqnarray} A_2&\ll&\frac{\phi^k(p-1)}{p^\frac{3}{2}}2^{k\omega(p-1)}\ln^{2k}p. \end{eqnarray}$

(3.7)

Inserting Eqs (3.3) and (3.7) into (3.1), we can deduce that

$\begin{eqnarray*} N(\mathcal{R}, p)& = &\frac{\phi^k(p-1)}{2^kp}+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^{\frac{k+1}{2}}}2^{k\omega(p-1)}\right)+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^\frac{3}{2}}2^{k\omega(p-1)}\ln^{2k}p\right)\\ & = &\frac{\phi^k(p-1)}{2^kp}+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^\frac{3}{2}}2^{k\omega(p-1)}\ln^{2k}p\right). \end{eqnarray*}$

This proves the Theorem 1.

Now we prove the Theorem 2. Let $\mathcal{A}$ denote the set of integers $1\leq x\leq p$ such that

$\begin{eqnarray*} \prod\limits_{i = 1}^{k}f(x+c_i)\equiv 0 (\bmod p). \end{eqnarray*}$

By the definition of primitive roots and Lehmer number, it follows that

$\begin{eqnarray} &&M(f(x), \mathcal{R}, p)\\ & = &\frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_i}(f(x+c_i))\left(1-(-1)^{f(x+c_i)+\overline{f(x+c_i)}}\right)\right)\\ & = &\frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_i}(f(x+c_i))\right)\\ &&+\frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^k}\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_i}(f(x+c_i))\right) \sum\limits_{t = 1}^{k}(-1)^{t}\mathop{\sum\limits_{i_1 = 1}^{k}\cdots \sum\limits_{i_t = 1}^{k}}_{i_1 < \cdots < i_t}g_{i_1}\cdots g_{i_t}\\ & = &\frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^k}\left(B_1+B_2\right), \end{eqnarray}$

(3.8)

where $g_i = (-1)^{f(x+c_i)+\overline{f(x+c_i)}}, i = 1, 2, \ldots, k$ .

$\begin{eqnarray*} &&B_1 = \prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_i}(f(x+c_i))\right)\\ & = &\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}}1 +\prod\limits_{i = 1}^{k}\left(\mathop{\sum\limits_{d_i\mid p-1}}_{\prod\limits_{i = 1}^{k}d_i > 1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_i}(f(x+c_i))\right). \end{eqnarray*}$

Obviously,

$\begin{eqnarray*} \left|\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}}1-p\right|\leq kl. \end{eqnarray*}$

From Lemma 6 we have

$\begin{eqnarray*} &&\sum\limits_{\chi_{d_1}}\sum\limits_{\chi_{d_2}}\cdots \sum\limits_{\chi_{d_k}}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_1}(f(x+c_1)) \chi_{d_2}(f(x+c_2))\cdots \chi_{d_k}(f(x+c_k))\\ & = &\mathop{{\sum\limits_{s_1 = 1}^{d_1}}\ '}\cdots \mathop{{\sum\limits_{s_k = 1}^{d_k}}\ '}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi\left((f(x+c_1))^{\frac{s_1(p-1)}{d_1}}\cdots(f(x+c_k))^{\frac{s_k(p-1)}{d_k}}\right). \end{eqnarray*}$

Due to $d_1d_2\cdots d_k > 1$ , and

$\frac{s_i(p-1)}{d_i} < p-1\ \text{for}\ d_i > 1 (i = 1, 2, \ldots, k),$

from Lemma 4 we can get that the polynomial

$\begin{eqnarray*} (f(x+c_1))^{\frac{s_1(p-1)}{d_1}}\cdots(f(x+c_k))^{\frac{s_k(p-1)}{d_k}} \end{eqnarray*}$

has a root in $\bar{\mathbb{F}}_p$ with multiples less than $p-1$ , thus it can not be multiple of a $(p-1)$ -th power of polynomial over $\mathbb{F}_p$ . Take $g(x) = (f(x+c_1))^{\frac{s_1(p-1)}{d_1}}\cdots(f(x+c_k))^{\frac{s_k(p-1)}{d_k}}$ , in Lemma 2 we have

$\begin{eqnarray*} \left|\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}}\chi\left(f(x+c_1)^{\frac{s_1(p-1)}{d_1}}\cdots f(x+c_k)^{\frac{s_k(p-1)}{d_k}}\right)\right| < (kl-1)p^{\frac{1}{2}}. \end{eqnarray*}$

Hence, we have

$\begin{eqnarray} &&\left|B_1-(p-kl)\right| < (2^{k\omega(p-1)}-1)(kl-1)p^{\frac{1}{2}}\leq 2^{k\omega(p-1)}(kl-1)p^{\frac{1}{2}}. \end{eqnarray}$

(3.9)

Using the methods in the proof of Theorem 1 we have

$\begin{eqnarray*} &&g_i = \frac{1}{p^2}\sum\limits_{m_i, n_i = 1}^{p-1}e\left(\frac{m_i(f(x+c_i))+n_i\overline{f(x+c_i)}}{p}\right)U_{m_i}(u_i)U_{n_i}(v_i). \end{eqnarray*}$

From the above discussion and Lemma 5, we can obtain

$\begin{eqnarray} \left|B_2\right|& < &\left|\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\mu\left(d_i\right)}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_i}(f(x+c_i))\right) \sum\limits_{t = 1}^{k}(-1)^{t}\mathop{\sum\limits_{i_1 = 1}^{k}\sum\limits_{i_2 = 1}^{k}\cdots \sum\limits_{i_t = 1}^{k}}_{i_1 < i_2 < \cdots < i_t}g_{i_1}g_{i_2}\cdots g_{i_t}\right|\\ & < &\prod\limits_{i = 1}^{k}\left(\sum\limits_{d_i\mid p-1}\frac{\left|\mu\left(d_i\right)\right|}{\phi\left(d_i\right)}\sum\limits_{\chi_{d_i}}\right)\sum\limits_{t = 1}^{k}\binom{k}{t}T^{2t}_p\ln^{2t}p\cdot\left|\mathop{\sum\limits_{x = 1}^{p}}_{x\notin \mathcal{A}} \chi_{d_i}(f(x+c_i))\right.\\ &&\left.\cdot e\left(\frac{m_1(f(x+c_1))+n_1\overline{(f(x+c_1))}+\cdots+m_t(f(x+c_t))+n_t\overline{(f(x+c_t))}}{p}\right)\right|\\ & < & 2^{k\omega(p-1)}\cdot\sum\limits_{t = 1}^{k}\binom{k}{t}T^{2t}_p\ln^{2t}p(kl+l)p^{\frac{1}{2}}. \end{eqnarray}$

(3.10)

Combing Eqs (3.8), (3.9) and (3.10) we have

$\begin{eqnarray} &&\left|M(f(x), \mathcal{R}, p)-\frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^k}(p-kl)\right|\\ & < &\frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^k}\left[2^{k\omega(p-1)}(kl-1)p^{\frac{1}{2}}+2^{k\omega(p-1)}\cdot\sum\limits_{t = 1}^{k}\binom{k}{t}T^{2t}_p\ln^{2t}p(kl+l)p^{\frac{1}{2}}\right]\\ & = &\frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^k}2^{k\omega(p-1)}p^{\frac{1}{2}}\cdot\left[(kl-1)+((k+1)l)\sum\limits_{t = 1}^{k}\binom{k}{t}T^{2t}_p\ln^{2t}p\right]. \end{eqnarray}$

(3.11)

Then we have

$\begin{eqnarray*} &&M(f(x), \mathcal{R}, p) = \frac{1}{2^k}\frac{\phi^k(p-1)}{(p-1)^{k-1}}+\mathrm{O}\left(\frac{\phi^k(p-1)}{p^{k-\frac{1}{2}}}2^{k\omega(p-1)}\ln^{2k}p\right). \end{eqnarray*}$

This complete the proof of Theorem 2.

4. Conclusions

From two perspectives, this paper consider the distribution of $LPRs$ that are related to the generalized Golomb's conjecture. Theorem 1 extends the binary linear equation $ag_1+bg_2 = c$ to the multivariate linear equation $a_1g_1+a_2g_2+\cdots+a_kg_k = c$ , and uses the properties of Gauss sums to derive an asymptotic formula for the number of its solutions $g_1, g_2, \ldots, g_k$ that are $LPRs$ . Theorem 2 considers $k$ consecutive $LPRs$ and employs the upper bound estimation of the generalized Kloosterman sums to provide a more general result that for $f(x)\in {\mathbb{F}}_p[x]$ , $k$ polynomials $f(x+c_1), f(x+c_2), \ldots, f(x+c_k)$ are Lehmer primitive roots modulo $p$ .

Use of AI tools declaration

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

Acknowledgments

The author gratefully appreciates the referees and academic editor for their helpful and detailed comments.

This work is supported by the N. S. F. (12126357) of P. R. China and the Natural Science Basic Research Plan in Shaanxi Province of China (2023-JC-QN-0050).

Conflict of interest

The author declare there are no conflicts of interest.

References

[1]	G. Wang, G. Chen, Y. Chu, A new random subspace method incorporating sentiment and textual information for financial distress prediction, Electron. Commer. Res. Appl., 29 (2018), 30–49. https://doi.org/10.1016/j.elerap.2018.03.004 doi: 10.1016/j.elerap.2018.03.004
[2]	G. Wang, J. L. Ma, G. Chen, Y. Yang, Financial distress prediction: Regularized sparse-based random subspace with er aggregation rule incorporating textual disclosures, Appl. Soft Comput., 90 (2020), 106152. https://doi.org/10.1016/j.asoc.2020.106152 doi: 10.1016/j.asoc.2020.106152
[3]	Z. Halim, S. M. Shuhidan, Z. M. Sanusi, Corporation financial distress prediction with deep learning: analysis of public listed companies in malaysia, Bus. Process Manage. J., 27 (2021), 1163–1178. https://doi.org/10.1108/Bpmj-06-2020-0273. doi: 10.1108/Bpmj-06-2020-0273
[4]	P. du Jardin, A two-stage classification technique for bankruptcy prediction, Eur. J. Oper. Res., 254 (2016), 236–252. https://doi.org/10.1016/j.ejor.2016.03.008 doi: 10.1016/j.ejor.2016.03.008
[5]	D. Campa, M. D. Camacho-Minano, The impact of SME's pre-bankruptcy financial distress on earnings management tools, Int. Rev. Financ. Anal., 42 (2015), 222–234. https://doi.org/10.1016/j.irfa.2015.07.004 doi: 10.1016/j.irfa.2015.07.004
[6]	J. Bertomeu, E. Cheynel, E. Floyd, W. Pan, Using machine learning to detect misstatements, Rev. Accounting Stud., 26 (2021), 468–519. https://doi.org/10.1007/s11142-020-09563-8 doi: 10.1007/s11142-020-09563-8
[7]	J. Donovan, J. Jennings, K. Koharki, J. Lee, Measuring credit risk using qualitative disclosure, Rev. Accounting Stud., 26 (2021), 815–863. https://doi.org/10.1007/s11142-020-09575-4 doi: 10.1007/s11142-020-09575-4
[8]	W. Ben‐Amar, I. Belgacem, Do socially responsible firms provide more readable disclosures in annual reports, Corporate Social Responsib. Environ. Manage., 25 (2018), 1009–1018. https://doi.org/10.1002/csr.1517 doi: 10.1002/csr.1517
[9]	P. Hajek, R. Henriques, Mining corporate annual reports for intelligent detection of financial statement fraud–a comparative study of machine learning methods, Knowledge-Based Syst., 128 (2017), 139–152. https://doi.org/10.1016/j.knosys.2017.05.001 doi: 10.1016/j.knosys.2017.05.001
[10]	F. Mai, S. N. Tian, C. Lee, L. Ma, Deep learning models for bankruptcy prediction using textual disclosures, Eur. J. Oper. Res., 274 (2019), 743–758. https://doi.org/10.1016/j.ejor.2018.10.024 doi: 10.1016/j.ejor.2018.10.024
[11]	Y. B. Qian, A critical genre analysis of mda discourse in corporate annual reports, Discourse Commun., 14 (2020), 424–437. https://doi.org/10.1177/1750481320910525 doi: 10.1177/1750481320910525
[12]	W. H. Beaver, Financial ratios as predictors of failure, J. Accounting Res., 4 (1966), 71–111. https://doi.org/10.2307/2490171 doi: 10.2307/2490171
[13]	E. B. Deakin, A discriminant analysis of predictors of business failure, J. Accounting Res., 10 (1972), 167–179. https://doi.org/10.2307/2490225 doi: 10.2307/2490225
[14]	D. Carmichael, Auditor's Reporting Obligation: The Meaning and Implementation of the Fourth Standard of Reporting; Auditing Research Monographh, 1, American Institute of Certified Public Accountants, 1978.
[15]	M. E. Zmijewski, Methodological issues related to the estimation of financial distress prediction models, J. Accounting Res., 22 (1984), 59–82. https://doi.org/10.2307/2490859 doi: 10.2307/2490859
[16]	E. I. Altman, The Prediction of Corporate Bankruptcy: A Discriminant Analysis, University of California, Los Angeles, 1967.
[17]	A. I. Dimitras, S. H. Zanakis, C. Zopounidis, A survey of business failures with an emphasis on prediction methods and industrial applications, Eur. J. Oper. Res., 90 (1996), 487–513. https://doi.org/10.1016/0377-2217(95)00070-4 doi: 10.1016/0377-2217(95)00070-4
[18]	S. A. Ross, R. Westerfield, J. F. Jaffe, Corporate Finance, Irwin/McGraw-Hill, 1999.
[19]	Y. Ding, X. Song, Y. Zen, Forecasting financial condition of chinese listed companies based on support vector machine, Expert Syst. Appl., 34 (2008), 3081–3089. https://doi.org/10.1016/j.eswa.2007.06.037 doi: 10.1016/j.eswa.2007.06.037
[20]	R. B. Geng, I. Bose, X. Chen, Prediction of financial distress: An empirical study of listed chinese companies using data mining, Eur. J. Oper. Res., 241 (2015), 236–247. https://doi.org/10.1016/j.ejor.2014.08.016 doi: 10.1016/j.ejor.2014.08.016
[21]	S. Ruan, X. Sun, R. Yao, W. Li, Deep learning based on hierarchical self-attention for finance distress prediction incorporating text, Comput. Intell. Neurosci., 2021 (2021), 1165296. https://doi.org/10.1155/2021/1165296 doi: 10.1155/2021/1165296
[22]	F. Y. Lin, D. R. Liang, E. C. Chen, Financial ratio selection for business crisis prediction, Expert Syst. Appl., 38 (2011), 15094–15102. https://doi.org/10.1016/j.eswa.2011.05.035 doi: 10.1016/j.eswa.2011.05.035
[23]	E. I. Altman, Financial ratios, discriminant analysis and the prediction of corporate bankruptcy, J. Finance, 23 (1968), 589–609. https://doi.org/10.2307/2978933 doi: 10.2307/2978933
[24]	G. Wang, J. Ma, S. L. Yang, An improved boosting based on feature selection for corporate bankruptcy prediction, Expert Syst. Appl., 41 (2014), 2353–2361. https://doi.org/10.1016/j.eswa.2013.09.033 doi: 10.1016/j.eswa.2013.09.033
[25]	J. Huang, H. B. Wang, G. Kochenberger, Distressed chinese firm prediction with discretized data, Manage. Decis., 55 (2017), 786–807. https://doi.org/10.1108/Md-08-2016-0546 doi: 10.1108/Md-08-2016-0546
[26]	L. G. Zhou, K. P. Tam, H. Fujita, Predicting the listing status of chinese listed companies with multi-class classification models, Inf. Sci., 328 (2016), 222–236. https://doi.org/10.1016/j.ins.2015.08.036 doi: 10.1016/j.ins.2015.08.036
[27]	D. Alaminos, A. Del Castillo, M. A. Fernandez, A global model for bankruptcy prediction, PLoS One, 11 (2016), e0166693. https://doi.org/10.1371/journal.pone.0166693 doi: 10.1371/journal.pone.0166693
[28]	Y. P. Huang, M. F. Yen, A new perspective of performance comparison among machine learning algorithms for financial distress prediction, Appl. Soft Comput., 83 (2019), 105663. https://doi.org/10.1016/j.asoc.2019.105663 doi: 10.1016/j.asoc.2019.105663
[29]	K. Olorunnimbe, H. Viktor, Deep learning in the stock market-a systematic survey of practice, backtesting, and applications, Artif. Intell. Rev., 56 (2023), 2057–2109. https://doi.org/10.1007/s10462-022-10226-0 doi: 10.1007/s10462-022-10226-0
[30]	S. Ben Jabeur, V. Serret, Bankruptcy prediction using fuzzy convolutional neural networks, Res. Int. Bus. Finance, 64 (2023), 101844. https://doi.org/10.1016/j.ribaf.2022.101844 doi: 10.1016/j.ribaf.2022.101844
[31]	Y. D. Wang, Y. L. Jia, Y. H. Tian, J. Xiao, Deep reinforcement learning with the confusion-matrix-based dynamic reward function for customer credit scoring, Expert Syst. Appl., 200 (2022), 117013. https://doi.org/10.1016/j.eswa.2022.117013 doi: 10.1016/j.eswa.2022.117013
[32]	S. X. Li, W. X. Shi, J. C. Wang, H. S. Zhou, A deep learning-based approach to constructing a domain sentiment lexicon: A case study in financial distress prediction, Inf. Process. Manage., 58 (2021), 102673. https://doi.org/10.1016/j.ipm.2021.102673 doi: 10.1016/j.ipm.2021.102673
[33]	J. Jing, W. Yan, X. Deng, A hybrid model to estimate corporate default probabilities in china based on zero-price probability model and long short-term memory, Appl. Econ. Lett., 28 (2020), 413–420. https://doi.org/10.1080/13504851.2020.1757611 doi: 10.1080/13504851.2020.1757611
[34]	J. Sun, H. Li, H. Fujita, B. B. Fu, W. G. Ai, Class-imbalanced dynamic financial distress prediction based on adaboost-svm ensemble combined with smote and time weighting, Inf. Fusion, 54 (2020), 128–144. https://doi.org/10.1016/j.inffus.2019.07.006 doi: 10.1016/j.inffus.2019.07.006
[35]	X. D. Du, W. Li, S. M. Ruan, L. Li, Cus-heterogeneous ensemble-based financial distress prediction for imbalanced dataset with ensemble feature selection, Appl. Soft Comput., 97, (2020), 106758. https://doi.org/10.1016/j.asoc.2020.106758 doi: 10.1016/j.asoc.2020.106758
[36]	J. Sun, H. Fujita, Y. J. Zheng, W. G. Ai, Multi-class financial distress prediction based on support vector machines integrated with the decomposition and fusion methods, Inf. Sci., 559 (2021), 153–170. https://doi.org/10.1016/j.ins.2021.01.059 doi: 10.1016/j.ins.2021.01.059
[37]	H. Wang, X. Liu, Undersampling bankruptcy prediction: Taiwan bankruptcy data, PLoS One, 16 (2021), e0254030. https://doi.org/10.1371/journal.pone.0254030 doi: 10.1371/journal.pone.0254030
[38]	X. Wu, S. Du, An analysis on financial statement fraud detection for chinese listed companies using deep learning, IEEE Access, 10 (2022), 22516–22532. https://doi.org/10.1109/ACCESS.2022.3153478 doi: 10.1109/ACCESS.2022.3153478
[39]	J. Liu, J. Li, Risk analysis of textile industry foreign investment based on deep learning, Comput. Intell. Neurosci., 2022 (2022), 3769670. https://doi.org/10.1155/2022/3769670 doi: 10.1155/2022/3769670
[40]	P. Craja, A. Kim, S. Lessmann, Deep learning for detecting financial statement fraud, Decis. Support Syst., 139 (2020), 113421. https://doi.org/10.1016/j.dss.2020.113421 doi: 10.1016/j.dss.2020.113421
[41]	T. Mikolov, K. Chen, G. Corrado, J. Dean, Efficient estimation of word representations in vector space, arXiv preprint, (2013), arXiv: 1301.3781. https://doi.org/10.48550/arXiv.1301.3781
[42]	S. Hochreiter, J. Schmidhuber, Long short-term memory, Neural Comput., 9 (1997), 1735–80. https://doi.org/10.1162/neco.1997.9.8.1735 doi: 10.1162/neco.1997.9.8.1735
[43]	S. J. Yu, D. L. Liu, W. F. Zhu, Y. Zhang, S. M. Zhao, Attention-based lstm, gru and cnn for short text classification, J. Intell. Fuzzy Syst., 39 (2020), 333–340. https://doi.org/10.3233/Jifs-191171 doi: 10.3233/Jifs-191171
[44]	A. Galassi, M. Lippi, P. Torroni, Attention in natural language processing, IEEE Trans. Neural Networks Learn. Syst., 32 (2021), 4291–4308. https://doi.org/10.1109/TNNLS.2020.3019893 doi: 10.1109/TNNLS.2020.3019893
[45]	V. Mnih, N. Heess, A. Graves, Recurrent models of visual attention, arXiv preprint, (2014), arXiv: 1406.6247. https://doi.org/10.48550/arXiv.1406.6247
[46]	D. Bahdanau, K. Cho, Y. Bengio, Neural machine translation by jointly learning to align and translate, arXiv preprint, (2014), arXiv: 1409.0473. https://doi.org/10.48550/arXiv.1409.0473
[47]	Z. Yang, D. Yang, C. Dyer, X. He, A. Smola, E. Hovy, Hierarchical attention networks for document classification, in Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, ACL, San Diego, USA, (2016), 1480–1489.
[48]	J. Sun, H. Fujita, P. Chen, H. Li, Dynamic financial distress prediction with concept drift based on time weighting combined with adaboost support vector machine ensemble, Knowledge-Based Syst., 120 (2017), 4–14. https://doi.org/10.1016/j.knosys.2016.12.019 doi: 10.1016/j.knosys.2016.12.019
[49]	S. Zhao, K. Xu, Z. Wang, C. Liang, W. Lu, B. Chen, Financial distress prediction by combining sentiment tone features, Econ. Modell., 106 (2022), 105709. https://doi.org/10.1016/j.econmod.2021.105709 doi: 10.1016/j.econmod.2021.105709
[50]	J. B. Jing, W. W. Yan, X. M. Deng, A hybrid model to estimate corporate default probabilities in China based on zero-price probability model and long short-term memory, Appl. Econ. Lett., 28 (2021), 413–420. https://doi.org/10.1080/13504851.2020.1757611 doi: 10.1080/13504851.2020.1757611
[51]	Y. Chen, Convolutional Neural Network for Sentence Classification, Master's thesis, University of Waterloo, 2015.
[52]	A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, et al., Attention is all you need, arXiv preprint, (2017), arXiv: 1706.03762. https://doi.org/10.48550/arXiv.1706.03762

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Electronic Research Archive

1.1 1.7

Metrics

Article views(2338) PDF downloads(160) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Electronic Research Archive

A deep learning approach of financial distress recognition combining text

Related Papers:

Abstract

1. Introduction

2. Some lemmas

3. Proofs of the theorems

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Abstract

1. Introduction

2. Some lemmas

3. Proofs of the theorems

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

Electronic Research Archive

A deep learning approach of financial distress recognition combining text

Related Papers:

Abstract

1. Introduction

2. Some lemmas

3. Proofs of the theorems

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Abstract

1. Introduction

2. Some lemmas

3. Proofs of the theorems

4. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References