The main purpose of this paper is using the elementary methods and the number of the solutions of some congruence equations to study the calculating problem of the fourth power mean of one special two-term exponential sums, and give an exact calculating formula for it.
Citation: Wenpeng Zhang, Yuanyuan Meng. On the fourth power mean of one special two-term exponential sums[J]. AIMS Mathematics, 2023, 8(4): 8650-8660. doi: 10.3934/math.2023434
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Abstract
The main purpose of this paper is using the elementary methods and the number of the solutions of some congruence equations to study the calculating problem of the fourth power mean of one special two-term exponential sums, and give an exact calculating formula for it.
1.
Introduction
Let p be an odd prime. For any integers k>h≥1 and integers m and n with (n,p)=1, the two-term exponential sums S(m,n,k,h;p) is defined as
S(m,n,k,h;p)=p−1∑a=0e(mak+nahp),
where e(y)=e2πiy and i2=−1.
This sum play a very important role in the study of analytic number theory, so many number theorists and scholars had studied the various properties of S(m,n,k,h;p), and obtained a series of meaningful research results. For example, H. Zhang and W. P. Zhang [1] proved that for any odd prime p, one has
p−1∑m=1|p−1∑a=0e(ma3+nap)|4={2p3−p2 if 3∤p−1;2p3−7p2 if 3∣p−1,
(1.1)
where n represents any integer with (n,p)=1.
W. P. Zhang and D. Han [2] used elementary and analytic methods to obtain the identity:
p−1∑m=1|p−1∑a=0e(a3+map)|6=5p4−8p3−p2,
(1.2)
where p denotes an odd prime with 3∤(p−1).
Recently, W. P. Zhang and Y. Y. Meng [3] studied the sixth power mean of S(m,1,3,1;p), and proved that for any odd prime p, we have the identities
p−1∑m=1|p−1∑a=0e(ma3+ap)|6={5p3⋅(p−1), if p≡5mod6;p2⋅(5p2−23p−d2), ifp≡1mod6,
(1.3)
where 4p=d2+27⋅b2, and d is uniquely determined by d≡1mod3.
On the other hand, L. Chen and X. Wang [4] studied the fourth power mean of S(m,1,4,1;p), and proved that the identities
p−1∑m=1|p−1∑a=0e(ma4+ap)|4={2p2(p−2), if p≡7mod12,2p3, ifp≡11mod12,2p(p2−10p−2α2), ifp≡1mod24,2p(p2−4p−2α2), ifp≡5mod24,2p(p2−6p−2α2), ifp≡13mod24,2p(p2−8p−2α2), ifp≡17mod24,
(1.4)
where α=α(p)=p−12∑a=1(a+¯ap) is an integer satisfying the following identity (see Theorem 4–11 in [5]):
p=α2+β2=(p−12∑a=1(a+¯ap))2+(p−12∑a=1(a+r¯ap))2,
(∗p) denotes the Legendre's symbol, r is any quadratic non-residue modulo p and ¯a is the solution of the congruence equation a⋅x≡1modp.
In addition, T. T. Wang and W. P. Zhang [6] gave calculating formulae of the fourth and sixth power mean of S(m,n,2,1;p).
From the formulae (1.1)–(1.4), it is not difficult to see that the content of all these papers only involves h=1 in S(m,n,k,h;p). Through searching literature, we have not found papers dealing with the fourth power mean of the two-term exponential sums S(m,n,k,2;p) so far. Therefore, when k>h=2, it is difficult to obtain some ideal results.
In this paper, we use the elementary and analytic methods, and the number of the solutions of some congruence equations to study the calculating problem of the 2k-th power mean:
S2k(p)=p−1∑m=0|p−1∑a=0e(ma3+a2p)|2k,
and give an exact calculating formula for S4(p) with p≡3mod4.
That is, we give the following two conclusions:
Theorem 1.1.Let p be a prime with p≡11 mod 12, then we have the identity
To complete the proofs of our theorems, we need four simple lemmas. Of course, the proofs of these lemmas need some knowledge of elementary or analytic number theory, all these can be found in references [5] and [7,8], so we do not repeat them here. First, according to W. P. Zhang and J. Y. Hu [9] or B. C. Berndt and R. J. Evans [10], we have the following:
Lemma 2.1.Let p be an odd prime with p≡1 mod 3. Then for any third-order character λ modulo p, we have the identity
τ3(λ)+τ3(¯λ)=dp,
where τ(χ)=p−1∑a=1χ(a)e(ap) denotes the classical Gauss sums, 4p=d2+27⋅b2, and d is uniquely determined by d≡1 mod 3.
where (∗p) denotes the Legendre's symbol modulo p, and a⋅¯a≡1 mod p.
Proof. We only prove the second formula in Lemma 2.2. Similarly, we can deduce the first one. It is clear that from the properties of the complete residue system and the Legendre's symbol modulo p we have
Proof. We only prove the second formula in Lemma 2.3. Similarly, we can deduce the first one. If (3,p−1)=1, then from the properties of the complete residue system modulo p we have
It is clear that S4(p) is a real number. If p=12k+11, then 3∤(p−1), p≡3mod4, (−1p)=−1 and (3p)=1. Note that τ(χ2)=i⋅√p is a pure imaginary number. So from Lemma 2.2, Lemma 2.3, Lemma 2.4 and (3.1), we have
Proof. Similarly, if p=12k+7, then S4(p) is also a real number, note that 3∣(p−1), (−1p)=−1, (3p)=−1 and τ(χ2)=i⋅√p, so from Lemma 2.2, Lemma 2.3, Lemma 2.4 and (3.1) we have
From A. Weil's important works [14] and [15] we have the estimate:
|p−1∑a=1(a−1+¯ap)|=|p−1∑a=1(a3−a2+ap)|≪√p.
Combining this estimate and our theorems we can deduce the following:
Corollary 3.1.Let a be an integer, p−1≥a≥0. Let p be an odd prime with p≡3 mod 4. Then we have the asymptotic formula
p−1∑m=0|p−1∑a=0e(ma3+a2p)|4=2p3+O(p52).
Remark 3.1. In this paper, we only discussed the case p≡3 mod 4. If p≡1mod4, then we can not calculate the exact value of S4(p) yet. The reason is that we do not know the exact values of
The main results of this paper is to give an exact calculating formula for the fourth power mean of one special two-term exponential sums. That is,
S4(p)=p−1∑m=0|p−1∑a=0e(ma3+a2p)|4,
with the case p≡3mod4.
Here, we give an example to calculate the exact results of the prime number p satisfying conditions p≡7mod12 or p≡11mod12. The exact results of calculation are summarised in Table 1.
At the same time, our results also provides some new and effective method for the calculating problem of the fourth power mean of the higher order two-term exponential sums. We have reasons to believe that these works will play a positive role in promoting the study of relevant problems. Furthermore, it is still an open problem for the case of k>h≥3 for S(m,n,k,h;p), interested readers can continue this research.
Acknowledgments
The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper.
This work is supported by the N. S.F. (12126357) of P. R. China.
Conflict of interest
The authors declare no conflict of interest.
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Wenpeng Zhang, Yuanyuan Meng. On the fourth power mean of one special two-term exponential sums[J]. AIMS Mathematics, 2023, 8(4): 8650-8660. doi: 10.3934/math.2023434
Wenpeng Zhang, Yuanyuan Meng. On the fourth power mean of one special two-term exponential sums[J]. AIMS Mathematics, 2023, 8(4): 8650-8660. doi: 10.3934/math.2023434