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# On the Mean Value of High-Powers of a Special Character Sum Modulo a Prime

School of Mathematics, Northwest University, Xi’an, China

* Corresponding Author: Xiaodan Yuan. Email:

(This article belongs to the Special Issue: Application of Computer Tools in the Study of Mathematical Problems)

*Computer Modeling in Engineering & Sciences* **2023**, *136*(1), 943-953. https://doi.org/10.32604/cmes.2023.024363

**Received** 29 May 2022; **Accepted** 07 September 2022; **Issue published** 05 January 2023

## Abstract

In this paper, we use the elementary methods, the properties of Dirichlet character sums and the classical Gauss sums to study the estimation of the mean value of high-powers for a special character sum modulo a prime, and derive an exact computational formula. It can be conveniently programmed by the “Mathematica” software, by which we can get the exact results easily.## Keywords

Let p be an odd prime, the quadratic character modulo p is called the Legendre symbol, which is defined by

Many mathematicians have studied the properties of the Legendre symbol and obtained a series of important results (see [1–13]). Perhaps the most representative properties of the Legendre’s symbol are as follows:

Let p and q be two distinct odd primes, then one has the quadratic reciprocal formula (see [14]: Theorem 9.8 or [15]: Theorems 4–6)

For any odd prime p with

In fact, the integers

where s is any quadratic non-residue modulo p.

Now we consider a sum A(r) be similar to

In this paper, we give an exact computational formula for

Theorem. Let p be a prime with

where d and b are uniquely determined by

From this Theorem, we can immediately deduce the following four Corollaries:

Corollary 1. Let p be a prime with

Corollary 2. Let p be a prime with

Corollary 3. Let p be a prime with

Corollary 4. Let p be a prime with

Some notes: In our Theorem, we only discuss the case

Thus, for all prime p with

In addition, our Theorem holds for all negative integers.

Obviously, the advantage of our work is that it can transfer a complex mathematical computational problem into a simple form suitable for computer programming. It means that for any fixed prime p with

In this section, we give some simple Lemmas, which are necessary in the proofs of our Theorem. In addition, we need some properties of the classical Gauss sums and character sums, which can be found in many number theory books, such as [14,15] or [17], and we will not repeat them. First, we have the following:

Lemma 1. Let p be a prime with

where

Proof. See references [18] or [19].

Lemma 2. Let p be an odd prime, for any non-principal character

where

Proof. From the properties of the classical Gauss sums we have

On the other hand, for any integer b with

we also have

From Eqs. (2) and (3) we have the identity

or

This proves Lemma 2.

Lemma 3. Let p be a prime p with

Proof. From the characteristic function of the cubic residue modulo p, we have

Applying Eq. (4) we have

From the properties of the classical Gauss sums, we have

Taking

Note that

Similarly, we also have

Combining Eqs. (5), (8) and (9) we can deduce that

This proves Lemma 3.

Lemma 4. Let p be any odd prime with

where d is the same as defined in the Theorem.

Proof. Note that

Indeed, for any integer

This proves Lemma 4.

Lemma 5. Let p be any odd prime with

Proof. From the definition

and the orthogonality of characters modulo p, we have

From Eq. (10) we also have

If

Now Lemma 5 follows from Eqs. (11)–(14).

In this section, we complete the proof of our Theorem. It is clear that the characteristic equation of the third order linear recursive formula

is

Note that

It is clear that the three roots of Eq. (16) are

From Lemma 5 we have

Solving the Eq. (18) we can get

This proves our Theorem.

Obviously, using Lemma 4 we can also extend k in Lemma 5 to all negative integers, which leads to the Corollary 1 and the Corollary 2.

This completes the proofs of our all results.

In this paper, we give an exact computational formula for

where d and b are uniquely determined by

Meanwhile, the problems of calculating the mean value of high-powers of quadratic character sums modulo a prime are given.

In the end, we use the mathematical software “Mathematica” to program and calculate the exact values of

Acknowledgement: The authors would like to thank the editor and referees for their suggestions and critical comments that substantially improve the presentation of this work.

Funding Statement: This work was supported by the N. S. F. (12126357) of China.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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Appendix A.

Clear[b]

Clear[p];

Clear[a];

Clear[d];

Array[p, 20];

, ]

]

Array[d, 20];

, ]

]

S[pi, di, bi, ki]: = (1/3) ∗ (diki + ((−di + 9 ∗ bi)/2)ki + ((−di −9 ∗ bi)/2)ki)

If[Element[b, Integers],

],

]

]

]

## Cite This Article

**APA Style**

*Computer Modeling in Engineering & Sciences*,

*136*

*(1)*, 943-953. https://doi.org/10.32604/cmes.2023.024363

**Vancouver Style**

**IEEE Style**

*Comput. Model. Eng. Sci.*, vol. 136, no. 1, pp. 943-953. 2023. https://doi.org/10.32604/cmes.2023.024363