A note on the hybrid power mean involving the cubic Gauss sums and Kloosterman sums

1.   Introduction

Let $q$ be a positive integer. For any integers $m$ and $n$, The famous Kloosterman sums $K(m, n; q)$ is defined as follows:

where $e(y) = e^{2\pi i y}$, $i^2 = -1$ and $\overline{a}$ is defined by the congruence $a\cdot\overline{a}\equiv 1\bmod q$.

This sum plays a very important role in the study of analytic number theory, many number theory problems are closely related to it, so many scholars have studied the various properties about $K(m, n; q)$ and obtained a series of significant research results. For example, Kloosterman's pioneering work ^[3] proved the identity

where $p$ is an odd prime.

Estermann ^[4] studied the upper bound estimation of $K(m, n; q)$, and obtained the best estimation

where $(m, n, q)$ denotes the greatest common divisor of $m$, $n$ and $q$, $d(q)$ denotes the Dirichlet divisor function.

Zhang ^[5] used the elementary method to study the fourth power mean of $K(m, n;q)$ for general modulo $q$, and proved the identity

where $\omega(q)$ denotes the number of all distinct prime divisors of $q$, $\phi(q)$ denotes the Euler's function, and $\prod\limits_{p\parallel q}$ denotes the product over all prime divisors of $q$ such that $p\mid q$ and $p^2\nmid q$.

In addition, Chen and Hu ^[6] studied the hybrid power mean involving the cubic Gauss sums $A(m)$ and Kloosterman sums $K(m, n;p)$, i.e.,

and proved that for any positive integer $k\geq 1$, one has the third-order linear recurrence formula

with the first three values

where $4p = d^2+27\cdot b^2$, and $d$ is uniquely determined by $d\equiv 1\bmod 3$.

Some related contents can also be found in ^{[7,8,9,10,11,12,13,14,15,16]}, we would not list them all here.

Obviously, the result in ^[6] is meaningful, it gives a calculation method for all $S_k(p)$. However, this result does not look pretty, because it contains parameter $A(1)$. Therefore, we can not calculate the exact value of $S_k(p)$.

On the other hand, Kloosterman sums can be used to solve the Waring-Goldbach problem, to solve the problems of prime distribution over short intervals, mean estimation of the Riemann Zata function and Fourier coefficients in modular form, etc. Not only that, the properties of Kloosterman sums can also be used to determine the generalized Hamming weight or the weight distribution of some linear codes, so it has important applications in the field of communication, cryptography and coding theory. Therefore, it is necessary to further study the properties of Klooaterman sums.

Based on the above, in this paper, as a note of ^[6], we modify some parameters in ^[6] as follows:

then we will get some more concise and prettier results. That is, we will prove the following two conclusions:

Theorem 1. Let $p$ be a prime with $p\equiv 1\bmod 3$. Then for any integer $k$, we have the third-order linear recurrence formula

with the first three values $C_0(1, p) = 1$, $C_1(1, p) = dp$ and $C_2(1, p) = 2p\cdot \left(p+1\right)$, where $4p = d^2+27\cdot b^2$, and $d$ is uniquely determined by $d\equiv 1\bmod 3$.

Theorem 2. Let $p$ be a prime with $p\equiv 1\bmod 3$. Then for any integer $k$, we have the third-order linear recurrence formula

with the first three values

and

From these two theorems we may immediately deduce the following corollaries:

Corollary 1. For any prime $p$ with $p\equiv 1\bmod 3$, we have the identity

Corollary 2. For any prime $p$ with $p\equiv 1\bmod 3$, we have the identity

Corollary 3. For any prime $p$ with $p\equiv 1\bmod 3$, we have the identity

Some notes: In our theorems, we only consider the prime $p$ with $p\equiv 1\bmod 3$. In fact if $3\nmid (p-1)$, then for any integer $m$ with $(m, p) = 1$, we have $A\left(\overline{m}\right) = 0$ and $C_k(h, p) = 0$ for all $k\geq 1$. So in this case, the results are trivial.

Note that $K(m, n; p)$ is a real number, so if we replace $A\left(\overline{m}\right)$ in Theorem 2 with $A(m)$, then the values of $C_k(h, p)$ are the same as in ^[6].

In addition, since $A\left(\overline{m}\right)\neq 0$ for all $(m, p) = 1$, so the third-order linear recurrence formulas in Theorem 1 and Theorem 2 are also hold for all integers $k < 3$.

For any integer $h\geq 3$, whether there is an exact calculating formula for $C_k(h, p)$ is an open problem. It remains to be further studied.

2.   Several lemmas

To complete the proofs of our all results, we need three necessary lemmas. The proofs of these lemmas requires some knowledge of elementary or analytic number theory, all these can be found in references ^[1,2], so we do not repeat them here. First we have the following:

Lemma 1. Let $p$ be an odd prime with $p\equiv 1\bmod 3$. Then for any third-order character $\lambda$ modulo $p$, we have the identity

where $\tau(\chi) = \sum\limits_{a = 1}^{p-1}\chi(a)e\left(\frac{a}{p}\right)$ denotes the classical Gauss sums, $4p = d^2+27\cdot b^2$, and $d$ is uniquely determined by $d\equiv 1\bmod 3$.

Proof. The proof of this Lemma see Zhang and Hu ^[17] or Berndt and Evans ^[18].

Lemma 2. Let $p$ be a prime with $p\equiv 1\bmod 3$, then for any three-order character $\lambda$ modulo $p$, we have

and

Proof. Let $p$ be a prime with $p\equiv1\bmod 3$, let $\lambda$ be any third-order character modulo $p$. Then for any integer $m$ with $(m, p) = 1$, from the properties of the third-order character modulo $p$ we have the identity

From the properties of the classical Gauss sums we have

and

From (2.1)–(2.3) and Lemma 1, we have

Note that $\tau(\lambda)\cdot \tau\left(\overline{\lambda}\right) = p$, from (2.4) and $\lambda^2 = \overline{\lambda}$ we also have

From (2.2), (2.3), (2.5), (2.6) and Lemma 1, we have

Now Lemma 2 follows from (2.4) and (2.7).

Lemma 3. Let $p$ be a prime with $p\equiv 1\bmod 3$. Then we have the identities

and

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

Similarly, we also have

From (2.1), (2.8), (2.9) and Lemma 1, we have

Note that the identity

from (2.5), (2.8), (2.9), (2.11) and Lemma 1, we have the identity

Now Lemma 3 follows from (2.10) and (2.12).

3.   Proofs of the theorems

Applying Lemma 1 and Lemma 2 we can easily prove our theorems. First we prove Theorem 1. For any prime $p$ with $p\equiv 1\bmod 3$ and integer $m$ with $(m, p) = 1$, from (2.1) and Lemma 1 we have the identity

From (3.1) we can deduce that

From (3.2) we can deduce the third-order linear recurrence formula

Taking $h = 1$, from Lemma 2 we have $C_0(1, p) = 1$, $C_1(1, p) = dp$ and $C_2(1, p) = 2p\cdot \left(p+1\right)$. This proves Theorem 1.

Now we prove Theorem 2. Taking $h = 2$ in (3.3), from (2.11) and Lemma 3 we have

and

This proves Theorem 2.

From (3.1) we know that $A\left(\overline{m}\right)\neq 0$. So formula (3.3) also holds for all integers $k < 0$. Taking $k = 2$ in Theorem 1 we have

or

This proves Corollary 1.

Taking $k = 2$ in Theorem 2, note that

and

Therefore, we have

and

This completes the proofs of our all results.

4.   Conclusions

The main results of this paper is to give two theorems for the hybrid power mean involving the cubic Gauss sums and Kloosterman sums. In addition, we also obtained a third-order linear recurrence formula for $C_k(h, p)$ with $h = 1$ and $2$. That is, for any integer $k$, we have the three-order linear recurrence formula

with the exact values $C_0(h, p)$, $C_1(h, p)$ and $C_2(h, p)$, where $d$ is uniquely determined by $4p = d^2+27b^2$ and $d\equiv 1\bmod 3$.

At the same time, our results also provides an effective method for the study of the hybrid power mean involving the $k$-th Gauss sums and Kloosterman sums.

Acknowledgments

This work is supported by the N. S. F. (12126357) of China, the Doctoral Scientific Research Project of XAAI. The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.