On the two-term exponential sums and character sums of polynomials

Abstract The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of one kind hybrid power mean containing the character sums of polynomials and two-term exponential sums modulo p, an odd prime, and acquire several accurate asymptotic formulas for them.


Introduction
Let q ≥ be an integer and χ be a Dirichlet character modulo q. For any positive integers N and M with M > N, and rational coe cient polynomial f (x) of x with degree n, the character sums of polynomials mod q is de ned by S(χ, f ; q) = N+M a=N+ χ(f (a)).
It is well known that the upper bound estimate of S(χ, f ; q) is a particularly vital classical problem in analytic number theory. Any substantial progress in this area will certainly play a valuable role in promoting the development of analytic number theory. For this reason, a great number of scholars have researched the estimate problem of S(χ, f ; q), and obtained a series of meaningful results. For instance, Pólya and Vinogradov's ground breaking work (see [1]: Theorem 8.21 and Theorem 13.15) proved that for any nonprincipal character χ mod q, one has the estimate N+M a=N+ χ (a) q ln q, where the symbol A B denotes |A| < cB for some constant c. Suppose that q = p is an odd prime. A. Weil's [2] proved a particularly signi cant conclusion: Let χ be a q-th character mod p, and polynomial f (x) is not a perfect q-th power mod p, then the estimate can be acquired, where the estimate p in (1.1) is the best one. Actually, Zhang Wenpeng and Yi Yuan [3] gave a series of polynomials where (r − s, q) = , m, n and χ satisfy several special conditions. The minor term ln p in (1.1) is di cult to improve, and it cannot even be improved to ln λ p, where < λ < is any xed real number.
A lot of results associated with character sums of polynomials can be found in various analytic number theory books, such as [4,5], and several papers about character sums of polynomials can be found in [6][7][8][9][10]. We are not going to list them one by one.
On the other hand, for any integers m and n, the two-term exponential sums G(m, n, r, s; q) is de ned as where r > s ≥ are integers, and e(y) = e πiy . It is necessary to research two-term exponential sums. In fact, if r = p is an odd prime, they are closely related to Fourier analysis on nite elds. Because of this, a lot of researchers have discussed the various properties of G(m, n, r, s; q), and obtained a great number of meaningful results, see [11][12][13][14][15][16][17][18][19][20][21][22][23]. For instance, Zhang Wenpeng and Han Di [24] researched the sixth power mean of the two-term exponential sums, and obtained an exact computational formula. Zhang where p indicates an odd prime with (n, p) = .
In this paper, we are going to consider the computational problem of the hybrid power mean containing character sums of polynomials and two-term exponential sums H(r, s, t, χ; p) = Han Di [26] studied the asymptotic properties of the hybrid mean value involving the two-term exponential sums and polynomial character sums, and proved the following asymptotic formula where p is an odd prime, χ denotes any non-principal even Dirichlet character mod p, and a represents the multiplicative inverse of a mod p. That is, aa ≡ mod p.
If we take k = − in this theorem, one can deduce the asymptotic formula Du Xiaoying [27] researched a similar problem, and proved the following conclusion: Let p > be a prime with ( , p − ) = . Then for any non-principal even character χ mod p, one has the identity where * p denotes the Legendre symbol mod p. According to this formula, Du Xiaoying [27] deduced the following asymptotic formula: The main aim of this paper is to use the analytic methods and the properties of the classical Gauss sums to research the computational problem of (1.2) for special integers r = , s = and t = or . We will give several sharp asymptotic formulas for (1.2). That is, we will prove the following two main results: Theorem 1. Let p be an odd prime with p ≡ mod and χ be any Dirichlet character mod p. If χ ≠ χ and χ ≠ χ , then we acquire the asymptotic formula where the error term E(p) satis es the estimate E(p) ≤ · p . If χ ≠ χ and χ = χ , then we acquire the asymptotic formula where the error term E (p) satis es the estimate E(p) ≤ · p .
Theorem 2. Let p be an odd prime with p ≡ mod , χ be any Dirichlet character mod p. If χ ≠ χ and χ ≠ χ , then we obtain the asymptotic formula where the error term W(p) satis es the estimate W(p) ≤ · p . If χ ≠ χ and χ = χ , then we obtain the asymptotic formula where the error term W (p) satis es the estimate W (p) ≤ · p .
Some notes: Above all, if (p − ) in Theorem 1, then for any non-principal character χ mod p, if χ = χ , then we acquire the identity p− a= χ a + ma = .
Therefore, in this case, the result is trivial. That is, If χ ≠ χ , then we acquire the identity In this case, we acquire the identity Secondly, if | (p − ) and χ is not a third character mod p (that is, there is not any character χ mod p such that χ = χ ), then we obtain the identity p− a= χ a + ma = .
Therefore, we did not discuss this special case.
Third, our methods can also be applied to the general hybrid power mean H( , , k, χ; p) for all integers k ≥ . Actually, if | k, then the asymptotic formula for H( , , k, χ; p) is the same as in Theorem 1. If k, then the asymptotic formula for H( , , k, χ; p) is the same as in Theorem 2.
Finally, it is worthwhile to improve the error term in Theorem 2.

Some lemmas
In this part, rstly, we introduce several simple properties related to classical Gauss sums mod q, which is de ned as , where e(y) = e πiy .
The other properties of τ(χ) can also be found in a great number of analytic number theory text books, such as [1] or [4] and [5], here we will not repeat them. If χ is a third character mod p and χ ≠ χ , then we acquire the identity If χ is a third character mod p and χ = χ , then we acquire the identity where λ is a third-order character mod p. That is, λ ≠ χ and λ = χ .
Proof. If χ is not a third character mod p, then there exists an integer r such that r ≡ mod p and χ(r) ≠ . According to the properties of the reduced residue system mod p, we obtain where λ is any third-order character mod p.
According to the properties of Gauss sums and reduced residue system, we obtain If χ ≠ χ , we can use the identity τ(λ)τ λ = p.
If χ = χ , then we get This proves Lemma 2.

Lemma 3.
Let p be an odd prime, then we obtain the identity Proof. It is not di cult to prove above formula by the same method of proving Lemma 2, so we omit details of the proof.  Proof. Note that τ(λ)τ λ = p, and λ = λ, from the method of proving Lemma 4, we obain Now Lemma 5 follows from (2.7), (2.8) and the identity |τ(χ)| = √ p.

Lemma 6.
Let p be an odd prime with p ≡ mod , then for any non-principal third character χ mod p, we can obtain the estimate where λ is any third-order character mod p.

Conclusion
The main results of this paper are two theorems. We obtained two sharp asymptotic formulas for the hybrid power mean involving character sums of polynomials and two-term exponential sums. In addition to this, we also acquired the upper bound estimation of the error terms. These results profoundly reveal the law of the value distribution of the character sums of polynomials and two-term exponential sums, and it can also be used for reference in the research of similar problems.