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.

Let

Many mathematicians have studied the properties of the Legendre symbol and obtained a series of important results (see [

Let

For any odd prime

In fact, the integers

Now we consider a sum

In this paper, we give an exact computational formula for

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

Thus, for all prime

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

7 | 1 | 1 | 1, 2, 3, 4, 5, 6, 7, 8 | |

19 | 7 | 1 | 1, 2, 3, 4, 5, 6, 7, 8 | |

31 | 4 | 2 | 1, 2, 3, 4, 5, 6, 7, 8 | |

61 | 1 | 3 | 1, 2, 3, 4, 5, 6, 7, 8 | |

73 | 7 | 3 | 1, 2, 3, 4, 5, 6, 7, 8 | |

97 | 19 | 1 | 1, 2, 3, 4, 5, 6, 7, 8 | |

103 | 13 | 3 | 1, 2, 3, 4, 5, 6, 7, 8 | |

151 | 19 | 3 | 1, 2, 3, 4, 5, 6, 7, 8 | |

163 | 25 | 1 | 1, 2, 3, 4, 5, 6, 7, 8 | |

181 | 7 | 5 | 1, 2, 3, 4, 5, 6, 7, 8 | |

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 [

On the other hand, for any integer

From

This proves Lemma 2.

Applying

From the properties of the classical Gauss sums, we have

Taking

Note that

Similarly, we also have

Combining

This proves Lemma 3.

Indeed, for any integer

This proves Lemma 4.

From

If

Now Lemma 5 follows from

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

Note that

It is clear that the three roots of

From Lemma 5 we have

Solving the

This proves our Theorem.

Obviously, using Lemma 4 we can also extend

This completes the proofs of our all results.

In this paper, we give an exact computational formula for

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

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

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