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On the generalized exponential sums and their fourth power mean

  • Wencong Liu and Shushu Ning EMAIL logo
Published/Copyright: March 20, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

The main purpose of this article is to study the calculating problem of the fourth power mean of the two-term exponential sums and provide an accurate calculating formula for utilizing analytical methods and character sums’ properties. In the meantime, a result of the fourth power mean of Gauss sums is improved.

Keywords: two-term exponential sums; Gauss sums; fourth power mean; analytic method; calculating formula
MSC 2010: 11L03; 11L05

1 Introduction

Let q ≥ 3 be a positive integer. For any integers m and n , the generalized two-term exponential sums C ( m , n , k , χ ; q ) are defined by

C ( m , n , k , χ ; q ) = ∑ a = 1 q χ ( a ) e m a k + n a q ,

where χ denotes any Dirichlet character modulo q , e ( y ) = e 2 π i y and i 2 = − 1 .

The generalized two-term exponential sums become Gauss sums when either m or n in C ( m , n , k , χ ; q ) is zero. These sums are crucial to analytic number theory and have a tight relationship to well-known number theory problems like the Waring problem and Goldbach conjecture. Therefore, many academics have investigated the two-term exponential sums and have developed several significant results.

Up to now, many researchers have studied the calculation and estimation of the high-th power mean of C ( m , n , k , χ ; q ) . This article focuses on the calculation and estimation of the fourth power mean of C ( m , n , k , χ ; q ) , which can be roughly divided into two types. They are

∑ m = 1 p − 1 ∣ C ( m , n , k , χ ; p ) ∣ 4 and ∑ χ mod p ∣ C ( m , n , k , χ ; p ) ∣ 4 .

Duan and Zhang [1] studied the fourth power mean of C ( m , n , 3 , χ ; p ) and proved that for any prime p with 3 ∤ ( p − 1 ) , one has the identities

∑ m = 1 p − 1 ∣ C ( m , n , 3 , χ ; p ) ∣ 4 = 3 p 3 − 8 p 2 if χ = χ 2 , 2 p 3 − 7 p 2 if χ = χ 0 , 2 p 3 − 3 p 2 − 3 p − 1 otherwise ,

where χ 0 and χ 2 = ⋅ p denote principal character and Legendre symbol modulo p , respectively. Li and Xu [2] proved that for any odd prime p , there are identities

∑ m = 1 p − 1 ∣ C ( m , 1 , 2 , χ ; p ) ∣ 4 = p 3 − 3 p 2 + 2 − 1 p p 2 − p − 8 − 1 p p if χ = χ 0 , 2 p 3 − 3 p 2 if χ ( − 1 ) = − 1 , 2 p 3 − 4 − 1 p p 2 − 3 p 2 − p ∑ a = 1 p − 1 χ ( a + a ¯ ) 2 if χ = χ 0 and χ ( − 1 ) = 1 ,

where a ¯ denotes the inverse of a modulo p , that is, a ¯ satisfied x ⋅ a ≡ 1 mod p .

Zhang [3] investigated moments of generalized quadratic Gauss sums weighted by L -functions and obtained

1 p − 1 ∑ χ mod p ∣ C ( m , 0 , 2 , χ ; p ) ∣ 4 = 3 p 2 − 6 p − 1 + 4 p m p if p ≡ 1 mod 4 , 3 p 2 − 6 p − 1 if p ≡ 3 mod 4 ,

and if p ≡ 3 mod 4 , then

1 p − 1 ∑ χ mod p ∣ C ( m , 0 , 2 , χ ; p ) ∣ 6 = 10 p 3 − 25 p 2 − 4 p − 1 ,

where m is any integer with ( m , p ) = 1 .

Zhang and Liu [4] have studied the fourth power mean of C ( 1 , 0 , 3 , χ ; p ) with 3 ∣ ( p − 1 ) and obtained a complex but closed-from expression. Precisely, they proved the identity

∑ χ mod p ∣ C ( 1 , 0 , 3 , χ ; p ) ∣ 4 = 5 p 3 − 18 p 2 + 20 p + 1 + U 5 p + 5 p U − 5 U 3 − 4 U 2 + 4 U ,

where U = ∑ a = 1 p e a 3 p is a real constant.

In 2020, Bag and Barman [5] proved that for odd prime p and any integer m with ( m , p ) = 1 , there are asymptotic formulae

∑ χ mod p ∣ C ( m , 0 , 2 , χ ; p ) ∣ 6 = 10 p 4 + O ( p 7 2 ) ,

∑ χ mod p ∣ C ( m , 0 , 2 , χ ; p ) ∣ 8 = 35 p 5 + O ( p 9 2 ) .

Over the next 2 years, Bag et al. [6,7] proved not only the asymptotic formula for the tenth power mean of C ( m , 0 , 2 , χ ; p ) but also the asymptotic formula for any 2 l -th power mean of C ( m , 0 , 2 , χ ; p ) . The results are as follows:

∑ χ mod p ∣ C ( m , 0 , 2 , χ ; p ) ∣ 10 = 126 p 6 + O ( p 11 2 ) ,

∑ χ mod p ∣ C ( m , 0 , 2 , χ ; p ) ∣ 2 l = 2 l − 1 l ⋅ p l + 1 + O ( p 2 l + 1 2 ) .

In addition, there are a number of other forms of the high-th power mean of C ( m , n , k , χ ; p ) , and if the reader is interested can refer to [8,9]. To save space, all the results are not given in this study.

In this article, we devote ourselves to the problem of calculating the fourth power mean of C ( 1 , 1 , 2 , χ ; p ) and C ( 1 , 0 , 3 , χ ; p ) . There has been some research on the fourth power mean of C ( 1 , 1 , 2 , χ ; p ) , but the summation over all Dirichlet character χ modulo p in ∣ C ( 1 , 1 , 2 , χ ; p ) ∣ 4 has not been studied so far, for which we will present interesting calculating formulae. For the fourth power mean of C ( 1 , 0 , 3 , χ ; p ) , Zhang and Liu [4] have given identities; we improved their result and give an exact calculating formula.

Theorem 1

Let p be an odd prime. Then we have the identities

1 p − 1 ∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 2 + a p 4 = 2 p 2 − 6 p − 1 + 4 p sin π 2 p if p ≡ 1 mod 4 , 2 p 2 − 6 p − 1 + 4 p cos π 2 p if p ≡ 3 mod 4 .

Example

Let p = 3 . Then we have

1 2 ∑ χ mod 2 ∑ a = 1 2 χ ( a ) e a 2 + a 3 4 = 1 2 ∑ χ = χ 0 , χ 2 χ ( 1 ) e 1 2 + 1 3 + χ ( 2 ) e 2 2 + 2 3 4 = 1 2 1 2 − 3 i 2 4 + 1 2 − 3 2 − 3 i 2 4 = 1 2 + 9 2 = 5 ,

and the right side of the equation is 2 × 3 × 3 − 6 × 3 − 1 + 4 × 3 cos π 6 = 5 .

Theorem 2

Let p be a prime. Then we have the identities

1 p − 1 ∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 3 p 4 = p 2 − p − 1 if 3 ∤ ( p − 1 ) , 5 p 2 − 13 p − 1 3 + ( D − 4 ) A k 2 + ( 4 − p ) A k + 2 p ( 4 − D ) 3 if 3 ∣ ( p − 1 ) ,

where A k = ω k D p 2 + D p 2 2 − p 3 1 2 1 3 + ω − k D p 2 − D p 2 2 − p 3 1 2 1 3 depends on p , k = 0 , k = 1 , or k = 2 , ω = − 1 + 3 i 2 and D is uniquely determined by 4 p = D 2 + 27 B 2 with D ≡ 1 mod 3 .

Example

Let p = 31 ; then D = 4 , in this time we have

∑ χ mod 31 ∑ a = 1 30 χ ( a ) e a 3 31 4 = 44010 − 810 ⋅ 31 3 ( 2 + 3 3 i ) 1 3 + ( 2 − 3 3 i ) 1 3 .

Corollary 1

For any odd prime p, we have the asymptotic formula:

∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 2 + a p 4 = 2 p 3 − 8 p 2 + O ( p 3 2 ) .

Corollary 2

For any prime p with p ≡ 1 mod 3 , we have the asymptotic formula:

∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 3 p 4 = 5 3 ⋅ p 3 + O ( p 5 2 ) .

In addition, it is natural to ask whether the fourth power mean

∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 3 + a p 4

has an exact calculating formula? This is an open problem.

2 Several lemmas

In order to make the structure of the article concise and complete, we need to introduce four simple lemmas in this section. The proof of these lemmas requires some knowledge of elementary or analytic number theory, all of which can be found in [10] and [11], so we will not repeat them here.

Lemma 1

Let p be an odd prime. Then we have the identities

∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p = ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( b 2 − 1 ) − d ( b − 1 ) p = p + 1 − 2 p sin π 2 p if p ≡ 1 mod 4 , p + 1 − 2 p cos π 2 p if p ≡ 3 mod 4 .

Proof

From some properties of the classical Gauss sums and the reduced (complete) residue system modulo p , we have

(1) ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p = ∑ b = 0 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 + 2 b ) + d b p − ∑ d = 1 p − 1 e − d 2 − d p = p − 1 + ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( 1 + 2 b ¯ ) + d p − ∑ d = 1 p − 1 e − d 2 − d p = p − 1 + ∑ b = 0 p − 1 ∑ d = 1 p − 1 e d 2 ( 1 + 2 b ) + d p − ∑ d = 1 p − 1 e d 2 + d p − ∑ d = 1 p − 1 e − d 2 − d p = p − 1 − ∑ d = 1 p − 1 e d 2 + d p − ∑ d = 1 p − 1 e − d 2 − d p = p − 1 − ∑ d = 1 p − 1 e ( d + 2 ¯ ) 2 − 4 ¯ p − ∑ d = 1 p − 1 e − ( d + 2 ¯ ) 2 + 4 ¯ p = p − 1 − ∑ d = 0 p − 1 e d 2 − 4 ¯ p + 1 − ∑ d = 0 p − 1 e − d 2 + 4 ¯ p + 1 = p + 1 − e − 4 ¯ p ∑ d = 1 p − 1 d p e d p − e 4 ¯ p ∑ d = 1 p − 1 d p e − d p = p + 1 − e − 4 ¯ p τ ( χ 2 ) − χ 2 ( − 1 ) e 4 ¯ p τ ( χ 2 ) .

Note the identities

(2) τ ( χ 2 ) = p if p ≡ 1 mod 4 , i p if p ≡ 3 mod 4 ,

and

(3) χ 2 ( − 1 ) = 1 if p ≡ 1 mod 4 , − 1 if p ≡ 3 mod 4 .

Combining (1), (2), and (3), we have the identities

∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p = p + 1 − 2 p sin π 2 p if p ≡ 1 mod 4 , p + 1 − 2 p cos π 2 p if p ≡ 3 mod 4 .

In the same way, we also have

∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( b 2 − 1 ) − d ( b − 1 ) p = p + 1 − 2 p sin π 2 p if p ≡ 1 mod 4 , p + 1 − 2 p cos π 2 p if p ≡ 3 mod 4 .

This proves Lemma 1.□

Lemma 2

Let p be a prime with 3 ∣ ( p − 1 ) . Then for any third-order character λ modulo p, we have identity

(4) τ 3 ( λ ) + τ 3 ( λ ¯ ) = D p ,

where D is uniquely determined by 4 p = D 2 + 27 B 2 with D ≡ 1 mod 3 .

Proof

For detailed proof of Lemma 2, see [12] or [13].□

Lemma 3

Let p be a prime with 3 ∣ ( p − 1 ) , then for any third-order character λ modulo p, we have identity

τ 5 ( λ ) + τ 5 ( λ ¯ ) = D p ( τ ( λ ) + τ ( λ ¯ ) ) 2 − p 2 ( τ ( λ ) + τ ( λ ¯ ) ) − 2 D p 2 .

Proof

Using (4) and noting that τ ( λ ) τ ( λ ¯ ) = p , we obtain

(5) ( τ ( λ ) + τ ( λ ¯ ) ) 5 = ( τ 3 ( λ ) + τ 3 ( λ ¯ ) + 3 τ ( λ ) τ ( λ ¯ ) ( τ ( λ ) + τ ( λ ¯ ) ) ) ( τ ( λ ) + τ ( λ ¯ ) ) 2 = ( D p + 3 p ( τ ( λ ) + τ ( λ ¯ ) ) ) ( τ ( λ ) + τ ( λ ¯ ) ) 2 = D p ( τ ( λ ) + τ ( λ ¯ ) ) 2 + 3 p ( τ 3 ( λ ) + τ 3 ( λ ¯ ) + 3 τ ( λ ) τ ( λ ¯ ) ( τ ( λ ) + τ ( λ ¯ ) ) ) = D p ( τ ( λ ) + τ ( λ ¯ ) ) 2 + 3 p ( D p + 3 p ( τ ( λ ) + τ ( λ ¯ ) ) ) = D p ( τ ( λ ) + τ ( λ ¯ ) ) 2 + 9 p 2 ( τ ( λ ) + τ ( λ ¯ ) ) + 3 D p 2 .

Moreover, we have

(6) ( τ ( λ ) + τ ( λ ¯ ) ) 5 = τ 5 ( λ ) + τ 5 ( λ ¯ ) + 5 τ 4 ( λ ) τ ( λ ¯ ) + 10 τ 3 ( λ ) τ 2 ( λ ¯ ) + 10 τ 2 ( λ ) τ 3 ( λ ¯ ) + 5 τ ( λ ) τ 4 ( λ ¯ ) = τ 5 ( λ ) + τ 5 ( λ ¯ ) + 10 τ 2 ( λ ) τ 2 ( λ ¯ ) ( τ ( λ ) + τ ( λ ¯ ) ) + 5 τ ( λ ) τ ( λ ¯ ) ( τ 3 ( λ ) + τ 3 ( λ ¯ ) ) = τ 5 ( λ ) + τ 5 ( λ ¯ ) + 10 p 2 ( τ ( λ ) + τ ( λ ¯ ) ) + 5 D p 2 .

Combining (5) and (6), we have

τ 5 ( λ ) + τ 5 ( λ ¯ ) = D p ( τ ( λ ) + τ ( λ ¯ ) ) 2 − p 2 ( τ ( λ ) + τ ( λ ¯ ) ) − 2 D p 2 .

This proves Lemma 3.□

Lemma 4

Let p be a prime with 3 ∣ ( p − 1 ) . Then for any third-order character λ modulo p, we have identity

τ ( λ ) τ 2 ( λ ¯ ) τ ( λ ) − 2 2 + τ ( λ ¯ ) τ 2 ( λ ) τ ( λ ¯ ) − 2 2 = ( D − 4 ) A k 2 + ( 4 − p ) A k + 2 p ( 4 − D ) ,

where A k = ω k D p 2 + D p 2 2 − p 3 1 2 1 3 + ω − k D p 2 − D p 2 2 − p 3 1 2 1 3 depends on p , k = 0 , k = 1 , or k = 2 , ω = − 1 + 3 i 2 .

Proof

From (4) and noting that τ ( λ ) τ ( λ ¯ ) = p , we have

τ ( λ ) τ 2 ( λ ¯ ) τ ( λ ) − 2 2 + τ ( λ ¯ ) τ 2 ( λ ) τ ( λ ¯ ) − 2 2 = τ ( λ ) τ 3 ( λ ¯ ) p − 2 2 + τ ( λ ¯ ) τ 3 ( λ ) p − 2 2 = τ 5 ( λ ¯ ) p − 4 τ 2 ( λ ¯ ) + 4 τ ( λ ) + τ 5 ( λ ) p − 4 τ 2 ( λ ) + 4 τ ( λ ¯ ) = 1 p ( τ 5 ( λ ) + τ 5 ( λ ¯ ) ) − 4 ( τ 2 ( λ ) + τ 2 ( λ ¯ ) ) + 4 ( τ ( λ ¯ ) + τ ( λ ) ) = D ( τ ( λ ) + τ ( λ ¯ ) ) 2 + ( 4 − p ) ( τ ( λ ) + τ ( λ ¯ ) ) − 2 D p − 4 ( τ 2 ( λ ) + τ 2 ( λ ¯ ) ) = ( D − 4 ) ( τ ( λ ) + τ ( λ ¯ ) ) 2 + ( 4 − p ) ( τ ( λ ) + τ ( λ ¯ ) ) + 2 p ( 4 − D ) .

For convenience, write the real number A = τ ( λ ) + τ ( λ ¯ ) , we construct cubic equation A 3 − 3 p A − D p = 0 based on (4) and τ ( λ ) τ ( λ ¯ ) = p . According to Cardans formula (formula of roots of a cubic equation), we have A = A 1 , A 2 or A 3 , and

A 1 = D p 2 + D p 2 2 + ( − p ) 3 1 2 1 3 + D p 2 − D p 2 2 + ( − p ) 3 1 2 1 3 , A 2 = ω D p 2 + D p 2 2 + ( − p ) 3 1 2 1 3 + ω 2 D p 2 − D p 2 2 + ( − p ) 3 1 2 1 3 , A 3 = ω 2 D p 2 + D p 2 2 + ( − p ) 3 1 2 1 3 + ω D p 2 − D p 2 2 + ( − p ) 3 1 2 1 3 ,

where ω = − 1 + 3 i 2 . We can determine three real numbers A k ( k = 0 , 1 , 2 ) according to prime p . This proves Lemma 4.□

3 Proofs of the theorems

3.1 Proof of Theorem 1

First from the orthogonality of the characters modulo p we have

1 p − 1 ∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 2 + a p 4 = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 ∑ d = 1 p − 1 a b ≡ c d mod p e a 2 + b 2 − c 2 − d 2 + a + b − c − d p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 ∑ d = 1 p − 1 a b ≡ c mod p e d 2 ( a 2 + b 2 − c 2 − 1 ) + d ( a + b − c − 1 ) p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( a 2 + b 2 − a 2 b 2 − 1 ) + d ( a + b − a b − 1 ) p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( a 2 − 1 ) ( b 2 − 1 ) − d ( a − 1 ) ( b − 1 ) p = ∑ a = 0 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( a 2 + 2 a ) ( b 2 − 1 ) − d a ( b − 1 ) p − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p = ( p − 1 ) 2 + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( 1 + 2 a ¯ ) ( b 2 − 1 ) − d ( b − 1 ) p − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p = ( p − 1 ) 2 + ∑ a = 0 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 a ( b 2 − 1 ) − d ( b − 1 ) p − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( b 2 − 1 ) − d ( b − 1 ) p − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p = ( p − 1 ) 2 + p ( p − 1 ) − p − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( b 2 − 1 ) − d ( b − 1 ) p − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p = 2 p 2 − 4 p + 1 − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e − d 2 ( b 2 − 1 ) − d ( b − 1 ) p − ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 2 ( b 2 − 1 ) + d ( b − 1 ) p .

Now taking result of Lemma 1 into the above we have the identities

1 p − 1 ∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 2 + a p 4 = 2 p 2 − 6 p − 1 + 4 p sin π 2 p if p ≡ 1 mod 4 , 2 p 2 − 6 p − 1 + 4 p cos π 2 p if p ≡ 3 mod 4 .

This proves Theorem 1.

3.2 Proof of Theorem 2

Case 1. If 3 ∤ ( p − 1 ) , then 1 3 , 2 3 , … , ( p − 1 ) 3 form a reduced residue system modulo p . Therefore,

(7) 1 p − 1 ∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 3 p 4 = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 ∑ d = 1 p − 1 a b ≡ c d mod p e a 3 + b 3 − c 3 − d 3 p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 ∑ d = 1 p − 1 a b ≡ c mod p e d 3 ( a 3 + b 3 − c 3 − 1 ) p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 3 ( a 3 − 1 ) ( b 3 − 1 ) p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d ( a − 1 ) ( b − 1 ) p = 2 ( p − 1 ) 2 − ( p − 1 ) − ( p − 2 ) 2 = p 2 − p − 1 .

Case 2. If 3 ∣ ( p − 1 ) , by some properties of the Gauss sums and third-order character sums modulo p , we have

1 p − 1 ∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 3 p 4 = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 ∑ d = 1 p − 1 a b ≡ c d mod p e a 3 + b 3 − c 3 − d 3 p = 1 3 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ c = 1 p − 1 ∑ d = 1 p − 1 a b ≡ c mod p e d 3 ( a 3 + b 3 − c 3 − 1 ) p = 1 3 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 3 ( a 3 + b 3 − a 3 b 3 − 1 ) p = 1 3 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 e d 3 ( a 3 − 1 ) ( b 3 − 1 ) p = ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 p ∣ ( a 3 − 1 ) ( b 3 − 1 ) 1 + ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 p ∤ ( a 3 − 1 ) ( b 3 − 1 ) e d 3 ( a 3 − 1 ) ( b 3 − 1 ) p = 2 ( p − 1 ) 2 − 3 ( p − 1 ) + 1 3 ∑ a = 1 p − 1 ∑ b = 1 p − 1 ∑ d = 1 p − 1 p ∤ ( a 3 − 1 ) ( b 3 − 1 ) ( 1 + λ ( d ) + λ ¯ ( d ) ) e d ( a 3 − 1 ) ( b 3 − 1 ) p = 2 ( p − 1 ) 2 − 3 ( p − 1 ) + 1 3 ∑ a = 1 p − 1 ∑ b = 1 p − 1 p ∤ ( a 3 − 1 ) ( b 3 − 1 ) ( − 1 + τ ( λ ) λ ¯ ( a 3 − 1 ) λ ¯ ( b 3 − 1 ) + τ ( λ ¯ ) λ ( a 3 − 1 ) λ ( b 3 − 1 ) ) = 2 ( p − 1 ) 2 − 3 ( p − 1 ) − 1 3 ( p − 4 ) 2 + 1 3 τ ( λ ) ∑ a = 1 p − 1 ∑ b = 1 p − 1 λ ¯ ( a 3 − 1 ) λ ¯ ( b 3 − 1 ) + 1 3 τ ( λ ¯ ) ∑ a = 1 p − 1 ∑ b = 1 p − 1 λ ( a 3 − 1 ) λ ( b 3 − 1 ) = 5 p 2 − 13 p − 1 3 + 1 3 τ ( λ ) ∑ a = 1 p − 1 λ ¯ ( a 3 − 1 ) 2 + 1 3 τ ( λ ¯ ) ∑ a = 1 p − 1 λ ( a 3 − 1 ) 2 = 5 p 2 − 13 p − 1 3 + 1 3 τ ( λ ) ∑ a = 1 p − 1 ( 1 + λ ( a ) + λ ¯ ( a ) ) λ ¯ ( a − 1 ) 2 + 1 3 τ ( λ ¯ ) ∑ a = 1 p − 1 ( 1 + λ ( a ) + λ ¯ ( a ) ) λ ( a − 1 ) 2

= 5 p 2 − 13 p − 1 3 + 1 3 τ ( λ ) − 2 + ∑ a = 1 p − 1 λ ¯ ( a ) λ ¯ ( a − 1 ) 2 + 1 3 τ ( λ ¯ ) − 2 + ∑ a = 1 p − 1 λ ( a ) λ ( a − 1 ) 2 = 5 p 2 − 13 p − 1 3 + 1 3 τ ( λ ) − 2 + 1 τ ( λ ) ∑ b = 1 p − 1 λ ( b ) ∑ a = 1 p − 1 λ ¯ ( a ) e b ( a − 1 ) p 2 + 1 3 τ ( λ ¯ ) − 2 + 1 τ ( λ ¯ ) ∑ b = 1 p − 1 λ ¯ ( b ) ∑ a = 1 p − 1 λ ( a ) e b ( a − 1 ) p 2 = 5 p 2 − 13 p − 1 3 + 1 3 τ ( λ ) τ 2 ( λ ¯ ) τ ( λ ) − 2 2 + 1 3 τ ( λ ¯ ) τ 2 ( λ ) τ ( λ ¯ ) − 2 2 .

Now taking result of Lemma 4 into the above we have the identity

(8) 1 p − 1 ∑ χ mod p ∑ a = 1 p − 1 χ ( a ) e a 3 p 4 = 5 p 2 − 13 p − 1 3 + ( D − 4 ) A k 2 + ( 4 − p ) A k + 2 p ( 4 − D ) 3 .

Combining (7) and (8), we complete the proof of Theorem 2.

Acknowledgements

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

  1. Funding information: This work was supported by the N. S. F. (12126357) of P. R. China.

  2. Author contributions: All authors contributed equally to this work. All authors read and approved the final manuscript.

  3. Conflict of interest: The authors declare that there are no conflicts of interest regarding the publication of this article.

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Received: 2023-11-29
Revised: 2024-02-13
Accepted: 2024-02-20
Published Online: 2024-03-20

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  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
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  24. Regular Articles
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  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
  29. Local and global solvability for the Boussinesq system in Besov spaces
  30. Construction of 4 x 4 symmetric stochastic matrices with given spectra
  31. A conjecture of Mallows and Sloane with the universal denominator of Hilbert series
  32. The uniqueness of expression for generalized quadratic matrices
  33. On the generalized exponential sums and their fourth power mean
  34. Infinitely many solutions for Schrödinger equations with Hardy potential and Berestycki-Lions conditions
  35. Computing the determinant of a signed graph
  36. Two results on the value distribution of meromorphic functions
  37. Zariski topology on the secondary-like spectrum of a module
  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
  40. Strong convergence for weighted sums of (α, β)-mixing random variables and application to simple linear EV regression model
  41. On the distribution of powered numbers
  42. Almost periodic dynamics for a delayed differential neoclassical growth model with discontinuous control strategy
  43. A new distributionally robust reward-risk model for portfolio optimization
  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
https://doi.org/10.1515/math-2023-0187
Keywords for this article
two-term exponential sums; Gauss sums; fourth power mean; analytic method; calculating formula
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  3. Upper bounds for the global cyclicity index
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  5. On the multiplicative sum Zagreb index of molecular graphs
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  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
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  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
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  21. A comprehensive review of the recent numerical methods for solving FPDEs
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  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
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  26. The product of a quartic and a sextic number cannot be octic
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  32. The uniqueness of expression for generalized quadratic matrices
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  38. On deferred f-statistical convergence for double sequences
  39. About j-Noetherian rings
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  41. On the distribution of powered numbers
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  44. Asymptotic behavior of solutions of a viscoelastic Shear beam model with no rotary inertia: General and optimal decay results
  45. Silting modules over a class of Morita rings
  46. Non-oscillation of linear differential equations with coefficients containing powers of natural logarithm
  47. Mutually unbiased bases via complex projective trigonometry
  48. Hyers-Ulam stability of a nonlinear partial integro-differential equation of order three
  49. On second-order linear Stieltjes differential equations with non-constant coefficients
  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
  51. The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains
  52. On discrete inequalities for some classes of sequences
  53. Boundary value problems for integro-differential and singular higher-order differential equations
  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
  55. Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension
  56. Endpoint boundedness of toroidal pseudo-differential operators
  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
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  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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