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On the fourth power mean of the general Kloosterman sums
Journal of Number Theory 169 (2016) 315–326
Contents lists available at ScienceDirect
Journal of Number Theory www.elsevier.com/locate/jnt
On the fourth power mean of the general Kloosterman sums ✩ Zhang Wenpeng School of Mathematics, Northwest University, Xi’an, Shaanxi, PR China
a r t i c l e
i n f o
Article history: Received 31 December 2015 Received in revised form 14 May 2016 Accepted 14 May 2016 Available online 12 July 2016 Communicated by David Goss
a b s t r a c t The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of the fourth power mean of the general Kloosterman sums for any primitive character χ mod q, and give an exact computational formula for it. © 2016 Elsevier Inc. All rights reserved.
MSC: 11L03 11L05 Keywords: The general Kloosterman sums Gauss sums The fourth power mean Identity
1. Introduction Let q ≥ 3 be a positive integer. For any integers m and n, the classical Kloosterman sums K(m, n; q) and general Kloosterman sums K(m, n, χ; q) are defined as follows: K(m, n; q) =
✩
q ma + na e q a=1
This work is supported by the NSF (11371291) of PR China. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jnt.2016.05.018 0022-314X/© 2016 Elsevier Inc. All rights reserved.
W. Zhang / Journal of Number Theory 169 (2016) 315–326
316
and K(m, n, χ; q) =
q
χ(a)e
a=1
where
q
ma + na q
,
denotes the summation over all 1 ≤ a ≤ q such that (a, q) = 1, e(y) = e2πiy ,
a=1
and a denotes the multiplicative inverse of a mod q (that is, aa ≡ 1 mod q), χ denotes a Dirichlet character mod q. Concerning the various properties of K(m, n; q) and K(m, n, χ; q), many authors have studied them, and obtained several of results, see [2–15]. For example, T. Estermann [3] proved the upper bound 1
1
|K(m, n; q)| ≤ (m, n, q) 2 · d(q) · q 2 , where (m, n, q) denotes the greatest common divisor of the numbers m, n and q, and d(q) denotes the number of divisors of q. A.V. Malyshev [10] obtained estimate: 1
1
|K(m, n, χ; p)| (m, n, p) 2 p 2 + , where p is a prime and denotes any fixed positive number. On the other hand, some authors also studied the mean values of K(m, n; q) and K(m, n, χ; q), and obtained a series of important results. For example, H. D. Kloosterman [7] studied the fourth power mean of K(a, 1; p), and proved the identity p−1
K 4 (a, 1; p) = 2p3 − 3p2 − 3p − 1.
a=1
For general odd integer q ≥ 3 and (n, q) = 1, Zhang Wenpeng [15] proved the identity q m=1
| K(m, n; q) | = 4
q
| K(m, 1; q) | = 3 4
m=1
ω(q) 2
q φ(q)
2 pq
1 4 − − 3 3p 3p(p − 1)
where φ(q) is Euler function, ω(q) denotes all distinct prime divisors of q and
denotes
pq
the product of all prime divisors of q such that p | q and (p, q/p) = 1. If p > 5, then combining the results of [9] and [11] one can deduce the identity p−1 a=0
K 5 (a, 1; p) =
p 3
4p3 + (ap + 5) p2 + 4p,
,
W. Zhang / Journal of Number Theory 169 (2016) 315–326
where
∗ p
317
is the Legendre symbol, ap is an integer satisfying |ap | < 2p and ⎧ 2 ⎪ ⎨ 2p − 12u 2 ap = 4x − 2p ⎪ ⎩0
if p = 3u2 + 5v 2 , if p = x2 + 15y 2 , if p ≡ 7, 11, 13, 14 mod 15.
If p > 6, the from the work of [4] one can also deduce that p−1
K 6 (a, 1; p) = 5p4 − 10p3 − (bp + 9) p2 − 5p,
a=0 3
where bp is an integer |bp | < 2p 2 . Using the very important work of Katz [6] one can give the asymptotic formula p−1
K 2h (a, 1; p) ∼
a=0
2h h
1 ph+1 , as p → ∞. h+1
Zhang Wenpeng [14], Li Jianghua and Liu Yanni [8] also proved the identity p−1 m=1
p−1 4 p−1 ma + na |K(m, n, χ; p)| = χ(a)e p 4
⎧ 3 2 ⎪ ⎨ 2p − 3p − 3p − 1 = 3p3 − 8p2 ⎪ ⎩ p2 (2p − 7)
m=1 a=1
if χ is the principal character mod p, if χ is the Legendre symbol mod p, if χ is a non-real character mod p,
(1)
where (n, p) = 1. But it seems that the fourth power means of the general Kloosterman sums K(m, n, χ; q) with composite modulus q have not been studied. At least we are not aware of such work. In this paper, we shall use the properties of Gauss sums and the analytic method to study this problem, and given an interesting identity for it. That is, we shall prove the following: Theorem. Let q > 3 be an odd number with q = N M and (N, M ) = 1, the prime power αk 1 α2 decompositions N = q1 q2 · · · qs , M = pα 1 p2 · · · pk , n be any integer with (n, q) = 1, ∗ qi
denotes the Legendre symbol mod qi (i = 1, 2, . . . , s), χj denotes any non-real α
primitive character mod pj j with αj ≥ 1 (j = 1, 2, . . . , k). Then for the primitive char ∗ ∗ acter χ = q∗1 q2 · · · qs χ1 χ2 · · · χk mod q, we have the identity q q 4 5 5 ma + na 2 3 − α + 1 − , χ(a) e = q φ(q) q p−1 α p−1 m=1 a=1 p|N
where pα q denotes that pα | q and pα+1 q.
p M
W. Zhang / Journal of Number Theory 169 (2016) 315–326
318
From this theorem we may immediately deduce the following: Corollary 1. Let q > 3 be an odd number with the prime power decomposition q = αk 1 α2 pα 1 p2 · · · pk , n be any integer with (n, q) = 1, χi be any non-real primitive character i mod pα i (i = 1, 2, . . . , k). Then for the primitive character χ = χ1 χ2 · · · χk mod q, we have the identity q q 4 5 ma + na 2 . α + 1 − χ(a) e = q φ(q) q p−1 α m=1 a=1 p q
If q is a square-full number (q > 1, p | q if and only if p2 | q), then from the above theorem we can also deduce the following: Corollary 2. Let q > 2 be an odd square-full number, n be any integer with (n, q) = 1. Then for any primitive character χ mod q, we have the identity q q 4 5 ma + na 2 α+1− . χ(a) e = q φ(q) q p−1 α m=1 a=1 p q
2. Several lemmas In this section, we present several lemmas which are necessary for the proofs of our theorems. Hereinafter, we shall use many properties of Gauss sums, all of them can be found in references [1], here we only give the definition of the Gauss sums τ (χ) and its some most important properties: Let q > 2 be an integer, χ be any primitive character mod q, then p−1 a ma √ , = χ(m)τ (χ) and |τ (χ)| = p. τ (χ) = χ(a)e χ(a)e p p a=1 a=1 p−1
These properties are used in the proofs of all Lemma 1, Lemma 2 and Lemma 3. First we have the following: Lemma 1. Let p be an odd prime, q = pα with α ≥ 2, χ be any primitive character mod q. Then for any integer n with (n, q) = 1, we have the identity q q 2 2 ma + na 2α 1− . χ(a)e =p q p m=1 a=1
Proof. From the definition and properties of Gauss sums we have
W. Zhang / Journal of Number Theory 169 (2016) 315–326
319
q q q 2 q q ma + na m(a − b) + n(a − b) χ(a)e χ(ab) e = q q m=1 a=1
a=1 b=1
m=1
q 2 m(a − 1) = χ(a) e χ(a) e = q a=1 a=1 m=1 b=1 m=1 ⎛ α ⎞ α−1 p 2 pα p m(a − 1) m(a − 1) ⎠ . − = χ(a) ⎝ e e (2) pα pα−1 a=1 m=1 m=1 q
q
q
mb(a − 1) + nb(a − 1) q
q
Note that the trigonometric identity q nm q = e q 0 m=1
if (q, n) = q, if (q, n) = q.
(3)
From the properties of the reduced residue system mod pα and (2) we have q q 2 ma + na χ(a)e q m=1 a=1 ⎛ α ⎞ α−1 α−1 p 2 p−1 p p α−1 α−1
+ s − 1) + s − 1) m(tp m(tp ⎠ − = χ tpα−1 + s ⎝ e e pα pα−1 t=0 s=1 m=1 m=1
= φ2 (pα ) + p2(α−1)
p−1
χ tpα−1 + 1 t=1
= φ (p ) − p 2
α
2(α−1)
+p
2(α−1)
= φ (p ) − p 2
α
2(α−1)
=p
2α
p−1
χ tpα−1 + 1 t=0
2 1− p
.
This proves Lemma 1. Lemma 2. Let p be an odd prime, q = pα with α ≥ 2, n be any integer with (n, p) = 1, χ1 be a primitive character mod pα . Then for any primitive character χ mod pα , we have the identity q 2 2 q ma + na χ(m) χ1 (a)e q m=1 a=1 p3α if χ1 χ and χ1 χ are primitive characters mod pα , = 0 otherwise.
W. Zhang / Journal of Number Theory 169 (2016) 315–326
320
Proof. It is clear that if a pass through a reduced residue system mod p, then for any integer b with (b, p) = 1, ab also pass through a reduced residue system mod p. From these properties and the definition of Gauss sums, we have q m=1
=
q 2 ma + na χ(m) χ1 (a)e q a=1
q q
χ1 (ab)
a=1 b=1
=
=
q
a=1
b=1 (b,q)=1
χ1 (a)
χ1 (a)
a=1
=
q
q
q
χ1 (a)χ(a)
χ(m)e
q
χ(m)e
mb(a − 1) + nb(a − 1) q
m=1
m=1
= τ (χ)
χ(m)e
q
χ(b)
b=1
q
m(a − b) + n(a − b) q
m=1
a=1 2
χ(m)e
m=1
q
q
q
m(a − 1) + nb(a − 1) q
m(a − 1) q
q
χ(b)e
b=1
nb(1 − a) q
χ1 (a)χ(a)χ(a − 1)χ(n(1 − a))
a=1 2
= χ(−n)τ (χ)
q
χ1 (a + 1)χ(a + 1)χ2 (a).
(4)
a=1
Since q = pα is odd with α ≥ 2, χ is a primitive character mod q, so χ2 is also a primitive mod q. If χ1 χ and χ1 χ are two primitive characters mod q, then we have q
χ1 (a + 1)χ(a + 1)χ2 (a)
a=1
q q 1 b(a + 1) 2 = χ1 χ(b) χ (a)e τ (χ1 χ) q a=1 b=1
q 1 b ba 2 = χ1 χ(b)e χ (a)e τ (χ1 χ) q a=1 q b=1 2 q 2 τ χ τ χ τ (χ1 χ) b = χ1 (b)χ(b)e = . τ (χ1 χ) q τ (χ1 χ) q
b=1
If χ1 χ is the principal character mod q, then we have q a=1
α
2
χ1 (a + 1)χ(a + 1)χ (a) =
p a=1 (a+1,p)=1
χ2 (a)
(5)
W. Zhang / Journal of Number Theory 169 (2016) 315–326 α
=
p
χ (a) − 2
α−1 p
a=1
χ2 (rp − 1) = 0.
321
(6)
r=1
If χ1 χ is a primitive character mod pβ with 1 ≤ β < α, then we have p p 1 b(a + 1) 2 χ1 (a + 1)χ(a + 1)χ (a) = χ1 χ(b) χ (a)e τ (χ1 χ, pβ ) pβ a=1 a=1 β
q
α
2
b=1
β
1 = χ1 χ(b)e τ (χ1 χ, pβ ) p
b=1
b pβ
pα
2
χ (a)e
a=1
ba pβ
pβ
τ χ2 b χ2 pα−β = 0, = χ1 (b)χ(b)e β β τ (χ1 χ, p ) p
(7)
b=1
a . pβ a=1 If χ1 χ is a primitive character mod pα and χ1 χ is not a primitive character mod pα , then χ1 χ must be a character mod pα−1 . Then from the process of proving (5) we have
where τ χ, p
β
β
=
p
χ(a)e
q τ χ2 b χ1 (a + 1)χ(a + 1)χ (a) = χ1 (b)χ(b)e τ (χ χ) q 1 a=1 q
2
b=1
2
τ χ = τ (χ1 χ)
α−1 p−1 p
r=0 s=1
χ1 χ rpα−1 + s e
rpα−1 + s pα
pα−1 p−1 τ χ2 s r = 0. χ1 χ (s) e e = τ (χ1 χ) s=1 pα r=0 p
(8)
Combining (4)–(8) we may immediately deduce the identity q q 2 2 ma + na χ(m) χ1 (a)e q m=1 a=1 p3α if χ1 χ and χ1 χ are the primitive characters mod pα , = 0 otherwise. This proves Lemma 2. Lemma 3. Let p be an odd prime, q = pα with α ≥ 2, n be any integer with (n, p) = 1, χ1 be a primitive character mod pα . Then for any primitive character χ mod pβ with 1 ≤ β < α, we have the identity
W. Zhang / Journal of Number Theory 169 (2016) 315–326
322
q q 2 2 4α−2 ma + na , p χ(m) χ1 (a)e = 4α−β q , p m=1 a=1
if β = 1 and χ = χ2 ; otherwise,
∗ p
where χ2 =
denotes the Legendre symbol mod p.
Proof. From the process of proving (4) we have q m=1
q 2 ma + na χ(m) χ1 (a)e q a=1
= χ(−n)
q
χ1 (a)χ(a)
a=1
q
m(a − 1) q
χ(m)e
m=1
2 .
(9)
Since χ is a primitive character mod pβ and 1 ≤ β < α, it follows that χ is not a primitive character mod pα . Note that the identity (3), from the properties of reduced residue system mod q we have
q
χ (m) e
m=1
β
=
p t=1
χ(t)e
m(a − 1) q
t(a − 1) pα
=
=
pα−β −1
p
r=0
t=1 (t,p)=1
β
χ rpβ + t e
(rpβ + t)(a − 1) pα
pα−β −1 r(a − 1) e pα−β r=0
0,
pα−β τ χ, pβ χ (a − 1)/pα−β ,
if (pα−β , a − 1) = pα−β ; if (pα−β , a − 1) = pα−β .
(10)
Note that χ1 is a primitive character mod pα , so χ1 χ is also a primitive character mod pα . From (9) and (10), we have q m=1
q 2 ma + na χ(m) χ1 (a)e q
= χ(−n)
a=1
q
χ1 (a)χ(a)
a=1
= χ(−n)
β p −1
q
χ(m)e
m=1
m(a − 1) q
2
2
χ1 χ tpα−β + 1 pα−β τ χ, pβ χ (t)
t=0 −1
p
β τ χ, p χ1 χ tpα−β + 1 χ2 (t). β
= χ(−n)p
2(α−β) 2
t=0
(11)
W. Zhang / Journal of Number Theory 169 (2016) 315–326
323
If β = 1 and χ = χ2 is the Legendre symbol mod p, then from (11) we have q m=1
q 2 ma + na χ2 (m) χ1 (a)e q a=1
p p−1 1 b(tpα−1 + 1) τ (χ, p) χ1 (b) e τ (χ1 ) pα t=1 α
= χ(−n)p
2(α−1) 2
b=1 pα
= χ(−n)p2(α−1) τ 2 (χ, p)
1 χ1 (b)e τ (χ1 ) b=1
= −χ(−n)p
b pα
p−1 bt e p t=1
2(α−1) 2
τ (χ, p) .
(12)
If 1 ≤ β < α and χ is a primitive character mod pβ with χ = χ2 , then we have β p −1
χ1 χ tpα−β + 1 χ2 (t) =
t=1
p p −1 1 b(tpα−β + 1) 2 χ1 χ(b) χ (t)e τ (χ1 χ) pα t=1 α
β
b=1
α
1 χ1 χ(b)e τ (χ1 χ) b=1
τ χ2 , pβ τ (χ1 χ) = . τ (χ1 χ) p
=
p −1
β
b pα
χ2 (t)e
t=1
bt pβ
(13)
Combining (11), (12) and (13) we have the identity q q 2 2 4α−2 ma + na , p χ(m) χ1 (a)e = 4α−β q p , m=1 a=1
if β = 1 and χ = χ2 ; otherwise.
This proves Lemma 3. Lemma 4. Let n, k1 and k2 be integers such that (k1 , k2 ) = 1. Let χ1 and χ2 be characters modulo k1 and k2 , respectively. Then there exist two integers n1 and n2 such that |K(m, n, χ1 χ2 ; k1 k2 )| = |K(m, n1 , χ1 ; k1 )| · |K(m, n2 , χ2 ; k2 )| , where n ≡ n1 k22 + n2 k12 mod k1 k2 . Proof. See exercise 15 for chapter 8 of [1].
W. Zhang / Journal of Number Theory 169 (2016) 315–326
324
3. Proof of the theorem In this section, we shall complete the proof of our theorem. Let χ1 be a primitive character mod pα , from the orthogonality of the characters mod pα we have α pα 2 2 p ma + na χ(m) χ1 (a)e pα a=1 χ mod pα m=1
α
α
= φ (p )
p m=1
pα 4 ma + na χ0 (m) χ1 (a)e , pα
(14)
a=1
where χ0 is the principal character mod pα . Let χ1 be a fixed primitive mod pα , g be a primitive root mod pα , character a·r(n) g r(n) ≡ n mod pα , χ1 (n) = e φ(pα ) , then (a, p) = 1. Let A(a) denote the number of all primitive characters χ mod pα such that χ1 χ and χ1 χ are also primitive characters mod pα . Then we have φ(pα )
φ(pα )
A(a) =
1=
b=1 (p, b(b+a)(b−a))=1 φ(pα )
= φ (p ) − α
b=1 p|b
φ(pα )
b=1
1−
1
b=1 p|b(b+a)(b−a)
φ(pα )
1−
φ(pα )
1−
b=1 p|(b+a)
1
b=1 p|(b−a)
α−1 3 α = φ (p ) − 3φ p . = φ (p ) 1 − p α
From Lemma 1, Lemma 2, Lemma 3 and (15) we have α pα p 2 2 ma + na χ(m) χ1 (a)e pα a=1 χ mod pα m=1
α pα p 2 2 ma + na = χ0 (m) χ1 (a)e α p m=1 a=1 α pα p 2 2 α ∗ ma + na + χ(m) χ1 (a)e α p a=1 β=1 χ mod pβ m=1 2 α−1 2 1 = p4α 1 − + p4α−2 + (p − 3)p4α−1 + p4α−β φ pβ 1 − p p β=2
(15)
W. Zhang / Journal of Number Theory 169 (2016) 315–326
3 + p3α φ (pα ) 1 − p 5 . = p2α φ2 (pα ) α + 1 − p−1
325
(16)
Combining (14) and (16) we may immediately deduce the identity pα pα 4 ma + na 5 2α α . χ1 (a)e = p φ (p ) α + 1 − pα p−1 m=1 a=1
(17)
αk 1 α2 Now for q = pα 1 p2 · · · pk = N M with (N, M ) = 1, from (1), (17), Lemma 4 and the properties of the reduced residue system mod q we have the identity
⎛ α ⎞ i pi i pα q q 4 4 k i ma + na mi a + ni a ⎟ ⎜ χ(a) e χi (a)e = ⎝ i ⎠ q pα i m=1 a=1 i=1 m =1 a=1 i
2α 3 2 α = p φ (p ) α + 1 − 3p − 8p p|N 2
= q φ(q)
p|N
pα M
5 3− p−1
pα M
5 p−1
5 α+1− p−1
.
This completes the proof of our theorem. Acknowledgment The author would like to thank the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. S. Chowla, On Kloosterman’s sums, Norkse Vid. Selbsk. Fak. Frondheim 40 (1967) 70–72. T. Estermann, On Kloostermann’s sums, Mathematica 8 (1961) 83–86. K. Hulek, J. Spandaw, B. van Geemen, D. van Straten, The modularity of the Barth–Nieto quintic and its relatives, Adv. Geom. 1 (2001) 263–289. H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, 1997, pp. 61–63. Nicholas M. Katz, Estimates for nonsingular multiplicative character sums, Int. Math. Res. Not. 7 (2002) 333–349. H.D. Kloosterman, On the representation of numbers in the form ax2 + by 2 + cz 2 + dt2 , Acta Math. 49 (1926) 407–464. Li Jianghua, Liu Yanni, Some new identities involving Gauss sums and general Kloosterman sums, Acta Math. Sinica (Chin. Ser.) 56 (2013) 413–416.
R. Livn˙e, Motivic orthogonal two-dimension representations of Gal Q/Q , Israel J. Math. 92 (1995) 149–156.
326
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[10] A.V. Malyshev, A generalization of Kloosterman sums and their estimates (in Russian), Vestnik Leningrad Univ. 15 (1960) 59–75. [11] C. Perter, J. Top, M. van der Vlugt, The Hasse zeta-function of a K3 surface related to the number of words of weight 5 in the melas codes, J. Reine Angew. Math. 432 (1992) 151–176. [12] H. Salié, Uber die Kloostermanschen Summen S(u, v; q), Math. Z. 34 (1931) 91–109. [13] Zhang Wenpeng, On the general Kloosterman sums and its fourth power mean, J. Number Theory 104 (2004) 156–161. [14] Zhang Wenpeng, On the fourth power mean of the general Kloosterman sums, Indian J. Pure Appl. Math. 35 (2004) 237–242. [15] Zhang Wenpeng, The fourth and sixth power mean of the classical Kloosterman sums, J. Number Theory 131 (2011) 228–238.
Contents lists available at ScienceDirect
Journal of Number Theory www.elsevier.com/locate/jnt
On the fourth power mean of the general Kloosterman sums ✩ Zhang Wenpeng School of Mathematics, Northwest University, Xi’an, Shaanxi, PR China
a r t i c l e
i n f o
Article history: Received 31 December 2015 Received in revised form 14 May 2016 Accepted 14 May 2016 Available online 12 July 2016 Communicated by David Goss
a b s t r a c t The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of the fourth power mean of the general Kloosterman sums for any primitive character χ mod q, and give an exact computational formula for it. © 2016 Elsevier Inc. All rights reserved.
MSC: 11L03 11L05 Keywords: The general Kloosterman sums Gauss sums The fourth power mean Identity
1. Introduction Let q ≥ 3 be a positive integer. For any integers m and n, the classical Kloosterman sums K(m, n; q) and general Kloosterman sums K(m, n, χ; q) are defined as follows: K(m, n; q) =
✩
q ma + na e q a=1
This work is supported by the NSF (11371291) of PR China. E-mail address: [email protected].
http://dx.doi.org/10.1016/j.jnt.2016.05.018 0022-314X/© 2016 Elsevier Inc. All rights reserved.
W. Zhang / Journal of Number Theory 169 (2016) 315–326
316
and K(m, n, χ; q) =
q
χ(a)e
a=1
where
q
ma + na q
,
denotes the summation over all 1 ≤ a ≤ q such that (a, q) = 1, e(y) = e2πiy ,
a=1
and a denotes the multiplicative inverse of a mod q (that is, aa ≡ 1 mod q), χ denotes a Dirichlet character mod q. Concerning the various properties of K(m, n; q) and K(m, n, χ; q), many authors have studied them, and obtained several of results, see [2–15]. For example, T. Estermann [3] proved the upper bound 1
1
|K(m, n; q)| ≤ (m, n, q) 2 · d(q) · q 2 , where (m, n, q) denotes the greatest common divisor of the numbers m, n and q, and d(q) denotes the number of divisors of q. A.V. Malyshev [10] obtained estimate: 1
1
|K(m, n, χ; p)| (m, n, p) 2 p 2 + , where p is a prime and denotes any fixed positive number. On the other hand, some authors also studied the mean values of K(m, n; q) and K(m, n, χ; q), and obtained a series of important results. For example, H. D. Kloosterman [7] studied the fourth power mean of K(a, 1; p), and proved the identity p−1
K 4 (a, 1; p) = 2p3 − 3p2 − 3p − 1.
a=1
For general odd integer q ≥ 3 and (n, q) = 1, Zhang Wenpeng [15] proved the identity q m=1
| K(m, n; q) | = 4
q
| K(m, 1; q) | = 3 4
m=1
ω(q) 2
q φ(q)
2 pq
1 4 − − 3 3p 3p(p − 1)
where φ(q) is Euler function, ω(q) denotes all distinct prime divisors of q and
denotes
pq
the product of all prime divisors of q such that p | q and (p, q/p) = 1. If p > 5, then combining the results of [9] and [11] one can deduce the identity p−1 a=0
K 5 (a, 1; p) =
p 3
4p3 + (ap + 5) p2 + 4p,
,
W. Zhang / Journal of Number Theory 169 (2016) 315–326
where
∗ p
317
is the Legendre symbol, ap is an integer satisfying |ap | < 2p and ⎧ 2 ⎪ ⎨ 2p − 12u 2 ap = 4x − 2p ⎪ ⎩0
if p = 3u2 + 5v 2 , if p = x2 + 15y 2 , if p ≡ 7, 11, 13, 14 mod 15.
If p > 6, the from the work of [4] one can also deduce that p−1
K 6 (a, 1; p) = 5p4 − 10p3 − (bp + 9) p2 − 5p,
a=0 3
where bp is an integer |bp | < 2p 2 . Using the very important work of Katz [6] one can give the asymptotic formula p−1
K 2h (a, 1; p) ∼
a=0
2h h
1 ph+1 , as p → ∞. h+1
Zhang Wenpeng [14], Li Jianghua and Liu Yanni [8] also proved the identity p−1 m=1
p−1 4 p−1 ma + na |K(m, n, χ; p)| = χ(a)e p 4
⎧ 3 2 ⎪ ⎨ 2p − 3p − 3p − 1 = 3p3 − 8p2 ⎪ ⎩ p2 (2p − 7)
m=1 a=1
if χ is the principal character mod p, if χ is the Legendre symbol mod p, if χ is a non-real character mod p,
(1)
where (n, p) = 1. But it seems that the fourth power means of the general Kloosterman sums K(m, n, χ; q) with composite modulus q have not been studied. At least we are not aware of such work. In this paper, we shall use the properties of Gauss sums and the analytic method to study this problem, and given an interesting identity for it. That is, we shall prove the following: Theorem. Let q > 3 be an odd number with q = N M and (N, M ) = 1, the prime power αk 1 α2 decompositions N = q1 q2 · · · qs , M = pα 1 p2 · · · pk , n be any integer with (n, q) = 1, ∗ qi
denotes the Legendre symbol mod qi (i = 1, 2, . . . , s), χj denotes any non-real α
primitive character mod pj j with αj ≥ 1 (j = 1, 2, . . . , k). Then for the primitive char ∗ ∗ acter χ = q∗1 q2 · · · qs χ1 χ2 · · · χk mod q, we have the identity q q 4 5 5 ma + na 2 3 − α + 1 − , χ(a) e = q φ(q) q p−1 α p−1 m=1 a=1 p|N
where pα q denotes that pα | q and pα+1 q.
p M
W. Zhang / Journal of Number Theory 169 (2016) 315–326
318
From this theorem we may immediately deduce the following: Corollary 1. Let q > 3 be an odd number with the prime power decomposition q = αk 1 α2 pα 1 p2 · · · pk , n be any integer with (n, q) = 1, χi be any non-real primitive character i mod pα i (i = 1, 2, . . . , k). Then for the primitive character χ = χ1 χ2 · · · χk mod q, we have the identity q q 4 5 ma + na 2 . α + 1 − χ(a) e = q φ(q) q p−1 α m=1 a=1 p q
If q is a square-full number (q > 1, p | q if and only if p2 | q), then from the above theorem we can also deduce the following: Corollary 2. Let q > 2 be an odd square-full number, n be any integer with (n, q) = 1. Then for any primitive character χ mod q, we have the identity q q 4 5 ma + na 2 α+1− . χ(a) e = q φ(q) q p−1 α m=1 a=1 p q
2. Several lemmas In this section, we present several lemmas which are necessary for the proofs of our theorems. Hereinafter, we shall use many properties of Gauss sums, all of them can be found in references [1], here we only give the definition of the Gauss sums τ (χ) and its some most important properties: Let q > 2 be an integer, χ be any primitive character mod q, then p−1 a ma √ , = χ(m)τ (χ) and |τ (χ)| = p. τ (χ) = χ(a)e χ(a)e p p a=1 a=1 p−1
These properties are used in the proofs of all Lemma 1, Lemma 2 and Lemma 3. First we have the following: Lemma 1. Let p be an odd prime, q = pα with α ≥ 2, χ be any primitive character mod q. Then for any integer n with (n, q) = 1, we have the identity q q 2 2 ma + na 2α 1− . χ(a)e =p q p m=1 a=1
Proof. From the definition and properties of Gauss sums we have
W. Zhang / Journal of Number Theory 169 (2016) 315–326
319
q q q 2 q q ma + na m(a − b) + n(a − b) χ(a)e χ(ab) e = q q m=1 a=1
a=1 b=1
m=1
q 2 m(a − 1) = χ(a) e χ(a) e = q a=1 a=1 m=1 b=1 m=1 ⎛ α ⎞ α−1 p 2 pα p m(a − 1) m(a − 1) ⎠ . − = χ(a) ⎝ e e (2) pα pα−1 a=1 m=1 m=1 q
q
q
mb(a − 1) + nb(a − 1) q
q
Note that the trigonometric identity q nm q = e q 0 m=1
if (q, n) = q, if (q, n) = q.
(3)
From the properties of the reduced residue system mod pα and (2) we have q q 2 ma + na χ(a)e q m=1 a=1 ⎛ α ⎞ α−1 α−1 p 2 p−1 p p α−1 α−1
+ s − 1) + s − 1) m(tp m(tp ⎠ − = χ tpα−1 + s ⎝ e e pα pα−1 t=0 s=1 m=1 m=1
= φ2 (pα ) + p2(α−1)
p−1
χ tpα−1 + 1 t=1
= φ (p ) − p 2
α
2(α−1)
+p
2(α−1)
= φ (p ) − p 2
α
2(α−1)
=p
2α
p−1
χ tpα−1 + 1 t=0
2 1− p
.
This proves Lemma 1. Lemma 2. Let p be an odd prime, q = pα with α ≥ 2, n be any integer with (n, p) = 1, χ1 be a primitive character mod pα . Then for any primitive character χ mod pα , we have the identity q 2 2 q ma + na χ(m) χ1 (a)e q m=1 a=1 p3α if χ1 χ and χ1 χ are primitive characters mod pα , = 0 otherwise.
W. Zhang / Journal of Number Theory 169 (2016) 315–326
320
Proof. It is clear that if a pass through a reduced residue system mod p, then for any integer b with (b, p) = 1, ab also pass through a reduced residue system mod p. From these properties and the definition of Gauss sums, we have q m=1
=
q 2 ma + na χ(m) χ1 (a)e q a=1
q q
χ1 (ab)
a=1 b=1
=
=
q
a=1
b=1 (b,q)=1
χ1 (a)
χ1 (a)
a=1
=
q
q
q
χ1 (a)χ(a)
χ(m)e
q
χ(m)e
mb(a − 1) + nb(a − 1) q
m=1
m=1
= τ (χ)
χ(m)e
q
χ(b)
b=1
q
m(a − b) + n(a − b) q
m=1
a=1 2
χ(m)e
m=1
q
q
q
m(a − 1) + nb(a − 1) q
m(a − 1) q
q
χ(b)e
b=1
nb(1 − a) q
χ1 (a)χ(a)χ(a − 1)χ(n(1 − a))
a=1 2
= χ(−n)τ (χ)
q
χ1 (a + 1)χ(a + 1)χ2 (a).
(4)
a=1
Since q = pα is odd with α ≥ 2, χ is a primitive character mod q, so χ2 is also a primitive mod q. If χ1 χ and χ1 χ are two primitive characters mod q, then we have q
χ1 (a + 1)χ(a + 1)χ2 (a)
a=1
q q 1 b(a + 1) 2 = χ1 χ(b) χ (a)e τ (χ1 χ) q a=1 b=1
q 1 b ba 2 = χ1 χ(b)e χ (a)e τ (χ1 χ) q a=1 q b=1 2 q 2 τ χ τ χ τ (χ1 χ) b = χ1 (b)χ(b)e = . τ (χ1 χ) q τ (χ1 χ) q
b=1
If χ1 χ is the principal character mod q, then we have q a=1
α
2
χ1 (a + 1)χ(a + 1)χ (a) =
p a=1 (a+1,p)=1
χ2 (a)
(5)
W. Zhang / Journal of Number Theory 169 (2016) 315–326 α
=
p
χ (a) − 2
α−1 p
a=1
χ2 (rp − 1) = 0.
321
(6)
r=1
If χ1 χ is a primitive character mod pβ with 1 ≤ β < α, then we have p p 1 b(a + 1) 2 χ1 (a + 1)χ(a + 1)χ (a) = χ1 χ(b) χ (a)e τ (χ1 χ, pβ ) pβ a=1 a=1 β
q
α
2
b=1
β
1 = χ1 χ(b)e τ (χ1 χ, pβ ) p
b=1
b pβ
pα
2
χ (a)e
a=1
ba pβ
pβ
τ χ2 b χ2 pα−β = 0, = χ1 (b)χ(b)e β β τ (χ1 χ, p ) p
(7)
b=1
a . pβ a=1 If χ1 χ is a primitive character mod pα and χ1 χ is not a primitive character mod pα , then χ1 χ must be a character mod pα−1 . Then from the process of proving (5) we have
where τ χ, p
β
β
=
p
χ(a)e
q τ χ2 b χ1 (a + 1)χ(a + 1)χ (a) = χ1 (b)χ(b)e τ (χ χ) q 1 a=1 q
2
b=1
2
τ χ = τ (χ1 χ)
α−1 p−1 p
r=0 s=1
χ1 χ rpα−1 + s e
rpα−1 + s pα
pα−1 p−1 τ χ2 s r = 0. χ1 χ (s) e e = τ (χ1 χ) s=1 pα r=0 p
(8)
Combining (4)–(8) we may immediately deduce the identity q q 2 2 ma + na χ(m) χ1 (a)e q m=1 a=1 p3α if χ1 χ and χ1 χ are the primitive characters mod pα , = 0 otherwise. This proves Lemma 2. Lemma 3. Let p be an odd prime, q = pα with α ≥ 2, n be any integer with (n, p) = 1, χ1 be a primitive character mod pα . Then for any primitive character χ mod pβ with 1 ≤ β < α, we have the identity
W. Zhang / Journal of Number Theory 169 (2016) 315–326
322
q q 2 2 4α−2 ma + na , p χ(m) χ1 (a)e = 4α−β q , p m=1 a=1
if β = 1 and χ = χ2 ; otherwise,
∗ p
where χ2 =
denotes the Legendre symbol mod p.
Proof. From the process of proving (4) we have q m=1
q 2 ma + na χ(m) χ1 (a)e q a=1
= χ(−n)
q
χ1 (a)χ(a)
a=1
q
m(a − 1) q
χ(m)e
m=1
2 .
(9)
Since χ is a primitive character mod pβ and 1 ≤ β < α, it follows that χ is not a primitive character mod pα . Note that the identity (3), from the properties of reduced residue system mod q we have
q
χ (m) e
m=1
β
=
p t=1
χ(t)e
m(a − 1) q
t(a − 1) pα
=
=
pα−β −1
p
r=0
t=1 (t,p)=1
β
χ rpβ + t e
(rpβ + t)(a − 1) pα
pα−β −1 r(a − 1) e pα−β r=0
0,
pα−β τ χ, pβ χ (a − 1)/pα−β ,
if (pα−β , a − 1) = pα−β ; if (pα−β , a − 1) = pα−β .
(10)
Note that χ1 is a primitive character mod pα , so χ1 χ is also a primitive character mod pα . From (9) and (10), we have q m=1
q 2 ma + na χ(m) χ1 (a)e q
= χ(−n)
a=1
q
χ1 (a)χ(a)
a=1
= χ(−n)
β p −1
q
χ(m)e
m=1
m(a − 1) q
2
2
χ1 χ tpα−β + 1 pα−β τ χ, pβ χ (t)
t=0 −1
p
β τ χ, p χ1 χ tpα−β + 1 χ2 (t). β
= χ(−n)p
2(α−β) 2
t=0
(11)
W. Zhang / Journal of Number Theory 169 (2016) 315–326
323
If β = 1 and χ = χ2 is the Legendre symbol mod p, then from (11) we have q m=1
q 2 ma + na χ2 (m) χ1 (a)e q a=1
p p−1 1 b(tpα−1 + 1) τ (χ, p) χ1 (b) e τ (χ1 ) pα t=1 α
= χ(−n)p
2(α−1) 2
b=1 pα
= χ(−n)p2(α−1) τ 2 (χ, p)
1 χ1 (b)e τ (χ1 ) b=1
= −χ(−n)p
b pα
p−1 bt e p t=1
2(α−1) 2
τ (χ, p) .
(12)
If 1 ≤ β < α and χ is a primitive character mod pβ with χ = χ2 , then we have β p −1
χ1 χ tpα−β + 1 χ2 (t) =
t=1
p p −1 1 b(tpα−β + 1) 2 χ1 χ(b) χ (t)e τ (χ1 χ) pα t=1 α
β
b=1
α
1 χ1 χ(b)e τ (χ1 χ) b=1
τ χ2 , pβ τ (χ1 χ) = . τ (χ1 χ) p
=
p −1
β
b pα
χ2 (t)e
t=1
bt pβ
(13)
Combining (11), (12) and (13) we have the identity q q 2 2 4α−2 ma + na , p χ(m) χ1 (a)e = 4α−β q p , m=1 a=1
if β = 1 and χ = χ2 ; otherwise.
This proves Lemma 3. Lemma 4. Let n, k1 and k2 be integers such that (k1 , k2 ) = 1. Let χ1 and χ2 be characters modulo k1 and k2 , respectively. Then there exist two integers n1 and n2 such that |K(m, n, χ1 χ2 ; k1 k2 )| = |K(m, n1 , χ1 ; k1 )| · |K(m, n2 , χ2 ; k2 )| , where n ≡ n1 k22 + n2 k12 mod k1 k2 . Proof. See exercise 15 for chapter 8 of [1].
W. Zhang / Journal of Number Theory 169 (2016) 315–326
324
3. Proof of the theorem In this section, we shall complete the proof of our theorem. Let χ1 be a primitive character mod pα , from the orthogonality of the characters mod pα we have α pα 2 2 p ma + na χ(m) χ1 (a)e pα a=1 χ mod pα m=1
α
α
= φ (p )
p m=1
pα 4 ma + na χ0 (m) χ1 (a)e , pα
(14)
a=1
where χ0 is the principal character mod pα . Let χ1 be a fixed primitive mod pα , g be a primitive root mod pα , character a·r(n) g r(n) ≡ n mod pα , χ1 (n) = e φ(pα ) , then (a, p) = 1. Let A(a) denote the number of all primitive characters χ mod pα such that χ1 χ and χ1 χ are also primitive characters mod pα . Then we have φ(pα )
φ(pα )
A(a) =
1=
b=1 (p, b(b+a)(b−a))=1 φ(pα )
= φ (p ) − α
b=1 p|b
φ(pα )
b=1
1−
1
b=1 p|b(b+a)(b−a)
φ(pα )
1−
φ(pα )
1−
b=1 p|(b+a)
1
b=1 p|(b−a)
α−1 3 α = φ (p ) − 3φ p . = φ (p ) 1 − p α
From Lemma 1, Lemma 2, Lemma 3 and (15) we have α pα p 2 2 ma + na χ(m) χ1 (a)e pα a=1 χ mod pα m=1
α pα p 2 2 ma + na = χ0 (m) χ1 (a)e α p m=1 a=1 α pα p 2 2 α ∗ ma + na + χ(m) χ1 (a)e α p a=1 β=1 χ mod pβ m=1 2 α−1 2 1 = p4α 1 − + p4α−2 + (p − 3)p4α−1 + p4α−β φ pβ 1 − p p β=2
(15)
W. Zhang / Journal of Number Theory 169 (2016) 315–326
3 + p3α φ (pα ) 1 − p 5 . = p2α φ2 (pα ) α + 1 − p−1
325
(16)
Combining (14) and (16) we may immediately deduce the identity pα pα 4 ma + na 5 2α α . χ1 (a)e = p φ (p ) α + 1 − pα p−1 m=1 a=1
(17)
αk 1 α2 Now for q = pα 1 p2 · · · pk = N M with (N, M ) = 1, from (1), (17), Lemma 4 and the properties of the reduced residue system mod q we have the identity
⎛ α ⎞ i pi i pα q q 4 4 k i ma + na mi a + ni a ⎟ ⎜ χ(a) e χi (a)e = ⎝ i ⎠ q pα i m=1 a=1 i=1 m =1 a=1 i
2α 3 2 α = p φ (p ) α + 1 − 3p − 8p p|N 2
= q φ(q)
p|N
pα M
5 3− p−1
pα M
5 p−1
5 α+1− p−1
.
This completes the proof of our theorem. Acknowledgment The author would like to thank the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976. S. Chowla, On Kloosterman’s sums, Norkse Vid. Selbsk. Fak. Frondheim 40 (1967) 70–72. T. Estermann, On Kloostermann’s sums, Mathematica 8 (1961) 83–86. K. Hulek, J. Spandaw, B. van Geemen, D. van Straten, The modularity of the Barth–Nieto quintic and its relatives, Adv. Geom. 1 (2001) 263–289. H. Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, 1997, pp. 61–63. Nicholas M. Katz, Estimates for nonsingular multiplicative character sums, Int. Math. Res. Not. 7 (2002) 333–349. H.D. Kloosterman, On the representation of numbers in the form ax2 + by 2 + cz 2 + dt2 , Acta Math. 49 (1926) 407–464. Li Jianghua, Liu Yanni, Some new identities involving Gauss sums and general Kloosterman sums, Acta Math. Sinica (Chin. Ser.) 56 (2013) 413–416.
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