The main purpose of this paper is to use the analytic methods and the properties of Gauss sums to study the computational problem of one kind of new sum analogous to Gauss sums and give an interesting fourth power mean and a sharp upper bound estimate for it.
1. Introduction
Let q≥3 be an integer, and let χ be a Dirichlet character modq. Then for any integer n, famous Gauss sum G(χ,n) is defined as follows:
(1)G(χ,n)=∑a=1qχ(a)·e(naq),
where e(y)=e2πiy.
This sum plays a very important role in the study of analytic number theory; many famous number theoretic problems are closely related to it. For example, the distribution of primes, Goldbach problem, the estimate of character sums, and the properties of Dirichlet L-functions are some good examples.
It is clear that if χ is a primitive Dirichlet character modq, then we have G(χ,n)=χ¯(n)·G(χ,1)≡χ¯(n)·τ(χ) and |τ(χ)|=q. Many properties of G(χ,n) and τ(χ) can be found in [1–3].
In this paper, we introduce a new sum analogous to Gauss sums as follows:
(2)G(χ,b,c,m;q)=∑a=0q-1χ(a2+ba+c)·e(maq),
where χ is a character modq, and b, c, and m are any integers with (m,q)=1.
The main purpose of this paper is to study the properties of G(χ,b,c,m;q), such as the following two problems:
giving an upper bound estimate of G(χ,b,c,m;q);
the problem of whether there exists a computational formula for the 2kth power mean
(3)∑c=0q-1|∑a=0q-1χ(a2+ba+c)·e(maq)|2k,k≥2.
It seems that no one has studied these problems yet; at least we have not seen any related results in the existing literature. The problems are interesting, because there exists a close relationship between the sum G(χ,b,c,m;q) and the generalized Kloosterman sums; they can also help us to further understand the properties of hybrid mean value of an exponential sum and character of a polynomial.
For general integer q>3, these two problems seem to be hard to make progress. But if q=p>2 is a prime and k=2, then we can prove the following two conclusions.
Theorem 1.
Let p be an odd prime and χ any nonprincipal character modp. Then for any integers b, c, and m with (m,p)=1, one has the estimate
(4)|∑a=0p-1χ(a2+ba+c)·e(map)|≤2p.
Theorem 2.
Let p be an odd prime and χ any nonprincipal character modp. Then for any integers b and m with (m,p)=1, one has the identity
(5)∑c=0p-1|∑a=0p-1χ(a2+ba+c)·e(map)|4={2p3-3p2-3p,ifχistheLegendresymbolmodp;2p3-6p2,ifχisanonrealcharactermodp.
Some Notes. If χ=χ0 is the principal character modp, then note that χ0(a2+ba+c)=0 or 1 and |G(χ0,b,c,m;p)|≤3. So, in this case, the result is trivial; we do not need to discuss problems (A) and (B).
For any integer k≥3, whether there exists an exact computational formula for the 2kth power mean
(6)∑c=0p-1|∑a=0p-1χ(a2+ba+c)·e(map)|2k,
is an open problem.
Furthermore, for general integer q>3, whether there exists a nontrivial upper bound estimate for G(χ,b,c,m;q) is also an interesting problem.
2. Several Lemmas
In this section, we will give several lemmas, which are necessary in the proof of our theorems. Hereinafter, we will use many properties of character sums and Gauss sums; all of these can be found in [1, 4]. So they will not be repeated here. First we have the following.
Lemma 3.
Let p be an odd prime; then, for any integer c with (c,p)=1, one has the identity
(7)∑a=0p-1e(ca2p)=(cp)·τ(χ2),
where χ2=(*/p) denotes the Legendre symbol.
Proof.
From the properties of Gauss sums and quadratic residue modp we have
(8)∑a=0p-1e(ca2p)=1+∑a=1p-1e(ca2p)=1+∑a=1p-1(1+(ap))·e(cap)=∑a=0p-1e(cap)+∑a=1p-1(ap)·e(cap)=(cp)∑a=1p-1(ap)·e(ap)=(cp)·τ(χ2).
This proves Lemma 3.
Lemma 4.
Let p be an odd prime and χ any nonprincipal Dirichlet character modp. Then for any integers b and c, one has the identity
(9)|∑a=0p-1χ(a2+ba+c)·e(ap)|=|∑r=1p-1χ¯χ2(r)·e((4c-b2)r-16¯·r¯p)|,
where r¯ denotes the solution of the congruence equation r·x≡1modp.
Proof.
Since χ is a nonprincipal Dirichlet character modp, from Lemma 3 and the properties of Gauss sums we have
(10)∑a=0p-1χ(a2+ba+c)·e(ap)=1τ(χ¯)·∑a=0p-1∑r=1p-1χ¯(r)e(r(a2+ba+c)p)·e(ap)=1τ(χ¯)·∑r=1p-1χ¯(r)×∑a=0p-1e(ra2+(br+1)a+crp)=1τ(χ¯)·∑r=1p-1χ¯(r)×∑a=0p-1e(4¯r(2a+r¯(br+1))2+cr-4¯r¯(br+1)2p)=1τ(χ¯)·∑r=1p-1χ¯(r)×∑u=0p-1e(4¯ru2+4¯·(4c-b2)r-2¯·b-4¯·r¯p)=1τ(χ¯)·∑r=1p-1χ¯(r)(4¯·rp)·τ(χ2)·e(4¯·(4c-b2)r-2¯·b-4¯·r¯p)=τ(χ2)τ(χ¯)·χ¯(4)·e(-2¯·bp)·∑r=1p-1χ¯(r)(rp)·e((4c-b2)r-16¯·r¯p).
For any nonprincipal character χmodp, we have |τ(χ)|=p. So, from (10) and noting that |χ¯(4)|=|e(-2¯·b/p)|=1, we have
(11)|∑a=0p-1χ(a2+ba+c)·e(ap)|=|∑r=1p-1χ¯χ2(r)·e((4c-b2)r-16¯·r¯p)|.
This proves Lemma 4.
Lemma 5.
Let p be an odd prime and χ any Dirichlet character modp. Then for any integers m and n, one has the estimate
(12)∑a=1p-1χ(a)·e(ma+na¯p)≤2p·(m,n,p)1/2,
where (m,n,p) denotes the greatest common divisor of m, n, and p.
Proof.
This estimate is by Weil [5], Chowla [6], Malyshev [7], and Estermann [8] with some minor modifications.
Lemma 6.
Let p be an odd prime; then, for any integer n with (n,p)=1, one has the calculating formula
(13)∑m=1p-1|∑a=1p-1χ(a)·e(ma+na¯p)|4={2p3-3p2-3p-1,ifχistheprincipalcharactermodp;3p3-8p2,ifχistheLegendre’ssymbolmodp;p2(2p-7),ifχisanonrealcharactermodp.
Proof.
See [9] or Corollary 2 of [10].
3. Proof of the Theorems
In this section, we will complete the proof of our theorems. First note that if a passes through a reduced residue system modp, then ma also passes through a reduced residue system modp, if (m,p)=1. From these properties we have
(14)|∑a=0p-1χ(a2+ba+c)·e(map)|=|∑a=0p-1χ(m¯2a2+m¯ba+c)·e(ap)|=|χ2(m¯)∑a=0p-1χ(a2+mba+cm2)·e(ap)|=|∑a=0p-1χ(a2+mba+cm2)·e(ap)|.
So without loss of generality we can assume that m=1. Now we prove Theorem 1. From Lemmas 4 and 5 we have
(15)|∑a=0p-1χ(a2+ba+c)·e(ap)|=|∑r=1p-1χ¯χ2(r)·e((4c-b2)r-16¯·r¯p)|≤2p·(4c-b2,16¯,p)1/2=2p.
This proves Theorem 1.
Now we prove Theorem 2. From the properties of a complete residue system modp we know that if c passes through a complete residue system modp, then 4c-b2 also passes through a complete residue system modp. So from Lemma 4 we have
(16)∑c=0p-1|∑a=0p-1χ(a2+ba+c)·e(ap)|4=∑c=0p-1|∑r=1p-1χ¯χ2(r)·e((4c-b2)r-16¯·r¯p)|4=∑m=0p-1|∑r=1p-1χ¯χ2(r)·e(mr-16¯·r¯p)|4=∑m=0p-1|∑r=1p-1χ¯χ2(r)·e(mr+r¯p)|4.
If χ=χ2 is the Legendre symbol, then χχ2=χ0 is the principal character modp, so from (16) and Lemma 6 we have
(17)∑c=0p-1|∑a=0p-1χ(a2+ba+c)·e(ap)|4=∑m=0p-1|∑r=1p-1χ¯χ2(r)·e(mr+r¯p)|4=1+∑m=1p-1|∑r=1p-1e(mr+r¯p)|4=1+2p3-3p2-3p-1=2p3-3p2-3p.
If χ is not a real character modp, then from (16) and Lemma 6 we have
(18)∑c=0p-1|∑a=0p-1χ(a2+ba+c)·e(ap)|4=∑m=0p-1|∑r=1p-1χ¯χ2(r)·e(mr+r¯p)|4=|∑r=1p-1χ¯χ2(r)·e(r¯p)|4+∑m=1p-1|∑r=1p-1χ¯χ2(r)·e(mr+r¯p)|4=p2+p2(2p-7)=2p2(p-3).
Now combining (14), (17), and (18) we may immediately deduce the identity
(19)∑c=0p-1|∑a=0p-1χ(a2+ba+c)·e(map)|4={2p3-3p2-3p,ifχis the Legendre symbol modp;2p3-6p2,ifχis a nonreal character modp.
This completes the proof of Theorem 2.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the referee for the very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P.S.F. (2013JZ001) and N.S.F. (11371291) of China.
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