Generalized twisted Kloosterman sum over ℤ[i]
The twisted Kloosterman sums over Z were studied by V. Bykovsky, A.Vinogradov, N. Kuznetsov, R. W. Bruggeman, R. J. Miatello, I. Pacharoni, A. Knightly, and C. Li. In our paper, we obtain similar estimates for K χ (α, β; γ; q) over ℤ[i] and improve the estimates obtained for the sums of this kind wi...
Gespeichert in:
| Datum: | 2014 |
|---|---|
| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Інститут математики НАН України
2014
|
| Schriftenreihe: | Український математичний журнал |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/166068 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Generalized twisted Kloosterman sum over ℤ[i] / S. Varbanets // Український математичний журнал. — 2014. — Т. 66, № 5. — С. 609–618. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-166068 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1660682020-02-19T01:26:39Z Generalized twisted Kloosterman sum over ℤ[i] Varbanets, S. Статті The twisted Kloosterman sums over Z were studied by V. Bykovsky, A.Vinogradov, N. Kuznetsov, R. W. Bruggeman, R. J. Miatello, I. Pacharoni, A. Knightly, and C. Li. In our paper, we obtain similar estimates for K χ (α, β; γ; q) over ℤ[i] and improve the estimates obtained for the sums of this kind with Dirichlet character χ (mod q 1), where q 1 | q. Узагальнені суми Клостермана з характером над Z вивчали В. Биковський, А. Виноградов, М. Кузнєцов, А. Найти та С. Лі. У статті отримано аналогічні оцінки для Kχ(α,β;γ;q) над Z[i], а також уточнено оцінки таких сум з характером Діріхле χ(modq1), де q1|q. 2014 Article Generalized twisted Kloosterman sum over ℤ[i] / S. Varbanets // Український математичний журнал. — 2014. — Т. 66, № 5. — С. 609–618. — Бібліогр.: 15 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166068 511.19 en Український математичний журнал Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Статті Статті |
| spellingShingle |
Статті Статті Varbanets, S. Generalized twisted Kloosterman sum over ℤ[i] Український математичний журнал |
| description |
The twisted Kloosterman sums over Z were studied by V. Bykovsky, A.Vinogradov, N. Kuznetsov, R. W. Bruggeman, R. J. Miatello, I. Pacharoni, A. Knightly, and C. Li. In our paper, we obtain similar estimates for K χ (α, β; γ; q) over ℤ[i] and improve the estimates obtained for the sums of this kind with Dirichlet character χ (mod q 1), where q 1 | q. |
| format |
Article |
| author |
Varbanets, S. |
| author_facet |
Varbanets, S. |
| author_sort |
Varbanets, S. |
| title |
Generalized twisted Kloosterman sum over ℤ[i] |
| title_short |
Generalized twisted Kloosterman sum over ℤ[i] |
| title_full |
Generalized twisted Kloosterman sum over ℤ[i] |
| title_fullStr |
Generalized twisted Kloosterman sum over ℤ[i] |
| title_full_unstemmed |
Generalized twisted Kloosterman sum over ℤ[i] |
| title_sort |
generalized twisted kloosterman sum over ℤ[i] |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| topic_facet |
Статті |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/166068 |
| citation_txt |
Generalized twisted Kloosterman sum over ℤ[i] / S. Varbanets // Український математичний журнал. — 2014. — Т. 66, № 5. — С. 609–618. — Бібліогр.: 15 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT varbanetss generalizedtwistedkloostermansumoverzi |
| first_indexed |
2025-07-14T20:42:10Z |
| last_indexed |
2025-07-14T20:42:10Z |
| _version_ |
1837656420364845056 |
| fulltext |
UDC 511.19
S. Varbanets (Odessa Nat. Univ)
GENERALIZED TWISTED KLOOSTERMAN SUM OVER Z[i]
УЗАГАЛЬНЕНА ГIБРИДНА СУМА КЛОСТЕРМАНА НАД Z[i]
The twisted Kloosterman sums over Z were studied by V. Bykovsky, A. Vinogradov, N. Kuznetsov, R. W. Bruggeman,
R. J. Miatello, I. Pacharoni, A. Knightly, and C. Li. In our paper, we obtain similar estimates of Kχ(α, β; γ; q) over Z[i]
and improve estimates obtained for the sums of this kind with Dirichlet character χ (mod q1), where q1 | q.
Узагальненi суми Клостермана з характером над Z вивчали В. Биковський, А. Виноградов, М. Кузнєцов, А. Найтлi
та С. Лi. У статтi отримано аналогiчнi оцiнки для Kχ(α, β; γ; q) над Z[i], а також уточнено оцiнки таких сум з
характером Дiрiхле χ (mod q1), де q1 | q.
1. Introduction. The classic Kloosterman sums appeared first in the work of Kloosterman [8] in
connection with the representation of natural numbers by binary quadratic forms. The Kloosterman
sum is an exponential sum over a reduced residue system modulo q:
K(a, b; q) :=
q∑
x=1
(x,q)=1
e
2πi
ax+bx−1
q (a, b ∈ Z q > 1 is a positive integer)
here and in the seque x−1 denote the reciprocal to x modulo q, i.e., xx−1 ≡ 1(mod q).
By the relation for q = q1q2, (q1, q2) = 1,
K(a, b; q) = K(aq′2, bq
′
2; q1) ·K(aq′1, bq
′
1; q2)
follows that suffices to obtain the estimations K(a, b; q) only for a case q = pn, p be a prime, n ∈ N.
The greatest difficultly in an estimation of the Kloosterman sums provides the case q = p.
The estimation K(a, b; p) � p3/4 under a condition (a, b, p) = 1 was obtained in the named work
of Kloosterman, and then Davenport [5] improved on it up to � p2/3. A. Weil [14] proved the
Riemann hypothesis for algebraic curves of over finite field and obtained for K(a, b; p) the best
possible estimation� p1/2.
Davenport [5] studied the general Kloosterman sums over finite field with the multiplicative
character χ of this field
Kχ(a, b; p) =
∑
x∈F∗p
χ(x)e
2πi
ax+bx−1
p .
The sums containing simultaneously multiplicative and additive characters call twisted or hybrid
sums.
The further generalizations of the Kloosterman sums concerned with a substitution of a prime
field Fp on it a finite expansion Fq, q = pn, 1 < n ∈ N. The generalizations of the Kloosterman
sums concerned with theory of modular forms studied in the works Kuznetsov [10, 11], Bruggeman
[2], Deshoiller and Iwaniec [6], Proskurin [12], R. W. Bruggeman, R. J. Miatello, I. Pacharoni [1],
A. Knightly, and C. Li [9].
c© S. VARBANETS, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5 609
610 S. VARBANETS
Let consider the ring of the Gaussian integers Z[i]. For Gaussian integers α, β, γ we can define
the Kloosterman sum
K(α, β; γ) =
∑
x∈Z[i]
x (mod γ)
(x,γ)=1
exp
(
πi Sp
αx+ βx−1
γ
)
.
R. W. Bruggeman and Y. Motohashi [3] obtained the estimation
K(α, β; γ)� 2ν(γ)N(γ)1/2N((α, β, γ))1/2,
where ν(γ) is the number distinct prime divisors of γ; (α, β, γ) denotes the greatest common divisor
of α, β, γ.
In [13] we considered two type of generalized Kloosterman sums over Z[i]
Kχ(α, β; k; γ) =
∑
x (mod γ)
(x,γ)=1
χ(x) exp
(
πiSp
αxk + βx−1
k
γ
)
,
where α, β, γ ∈ Z[i], χ is multiplicative character modulo γ, and
K̃(α, β;h, q; k) =
∑
x,y∈Z[i]
x,y (mod γ)
N(xy)≡h (mod q)
eq
(
1
2
Sp(αxk + βyk)
)
,
where α, β ∈ Z[i], h, q ∈ N, (h, q) = 1.
We call K(α, β; k; γ, χ) the twisted power Kloosterman sum and K̃(α, β;h, q; k) call the norm
Kloosterman sum.
In this paper we obtain the estimations of generalized Kloosterman sum with a Dirichlet character
over the ring of the Gaussian integers which extend the A. Knightly, C. Li [9] results.
Remark 1.1. We denote G := Z[i] the ring of the Gaussian integers
G =
{
a+ bi
∣∣a, b,∈ Z, i2 = −1
}
.
For the designation of the Gaussian integers we shall use the Greek letters α, β, γ, ξ, η; a Gaussian
prime number denote through p if p 6∈ Z. For α ∈ Z[i] we put Sp(α) = α + α = 2<(α), N(α) =
= α ·α, where α denotes a complex conjugate with α; Sp(α) and N(α) we name a trace and a norm
(respectively) of α from Q(i) into Q.
The writing a ∈ Zq (respectively, α ∈ Gγ) under the sign Σ denotes that a ∈ Z (respectively,
α ∈ G) and a (respectively, α) runs a complete residue system modulo q (modulo γ). Analogous,
a ∈ Z∗q (respectively, α ∈ G∗γ) denotes a ∈ Z (respectively, α ∈ G) and runs a reduced residue
system modulo q (respectively, modulo γ).
The writing
∑
(U)
denotes that the summation runs over the region U which describes separately.
For A ∈ N (or α ∈ G) put νp(A) = a (or νp(α) = a) if pa‖A (or pa‖α). Moreover, exp (z) = ez,
eq(z) = e
2πi
z
q for q ∈ N; the Vinogradov symbol as in f(x)� g(x) means that f(x) = O(g(x)).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
GENERALIZED TWISTED KLOOSTERMAN SUM OVER Z[i] 611
2. Auxiliary results. For the proof of our main results the following lemmas are needed.
Lemma 2.1. Let f(x) ∈ Z[x], f(x) = a1x + a2x
2 + pλ3a3x
3 + . . . + pλkakx
k, λj > 0,
j = 3, . . . , k; (ai, p) = 1, i = 2, 3, . . . , k; p > 2 be a prime number. Then for m ∈ N we have∑
x∈Zpm
epm(f(x)) = ε(m)pm/2epm(F (a1, . . . , ak)), (2.1)
where F (x1, . . . , xk) ∈ Z[x1, . . . , xk], and, moreover,
F (a1, a2, . . . , ak) ≡ −a21(2a2)−1 (mod p),
ε(m) =
1 if m is even,
i
(p−1
2
)2
if m is odd.
Proof. Setting x = y + pm−1z, y ∈ {0, 1, . . . , pm−1 − 1}, z ∈ {0, 1, . . . , p− 1}, we obtain
S :=
∑
x∈Zpm
epm(f(x)) =
∑
y∈Zpm−1
∑
z∈Zp
epm(f(y) + pm−1zf ′(y)).
The sum over z gives zero if f ′(y) 6≡ 0 (mod p).
We have f ′(y) = a1 + 2a2y (mod p). Thus
S = epm(f(y0))p
∑
y∈Zpm−1
epm−2(g(y)),
where y0 ∈ Zp, a1 +2a2y0 ≡ 0 (mod p), g(y) =
f(y0 + py)− f(y0)
p2
= b1y+ b2y
2 +pµ3b3y
3 + . . .
. . .+ pµkbky
k, b1 ≡
a1 + 2a2y0
p
(mod p), b2 ≡ a2 (mod p), bj ≡ aj (mod p), µj > 1.
These considerations we continue further.
Thereby for m ≡ 0 (mod 2) we obtain
S = pm/2epm
(
f(y0) + p2g(y1) + . . .
)
. (2.2)
For n is odd we have
S = p
m−1
2 epm
(
f(y0) + p2g(y1 + . . .)
) ∑
x∈Zp
ep(b1x+ a2x
2) =
= p
m
2 i
(p−1
2
)2
epm
(
f(y0) + p2g(y1) + . . .+ pm−1b′1
)
. (2.3)
Take into account that f(y0) ≡ −a21(2a2)−1 (mod p), we prove Lemma 2.1.
Lemma 2.2. Let p be the Gaussian prime number, α1, . . . , αk ∈ G, (αj , p) = 1, j = 2, 3, . . . ;
λj be a positive integer, j = 3, . . . , k. Then the relations
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
612 S. VARBANETS
∑
ξ∈Gmp
exp
(
πiSp
(
α1ξ + α2pξ
2 + α3p
λ3ξ3 + . . .+ αkp
λkξk
pm
))
=
=
0 if α1 6≡ 0 (mod p), p 6= 1 + i,
e
πiSp
(F1(α1,...,αk)
pm
)
N(p)
m+1
2 if α1 ≡ 0 (mod p), p 6= 1 + i,
0 if α1 6= 0 (mod p2), p = 1 + i,
e
πiSp
(F2(α1,...,αk)
pm
)
if α1 ≡ 0 (mod p2), p = 1 + i,
(2.4)
hold, where the polynomials F1, F2 are similar to F from Lemma 2.1.
This assertion can be proved exactly in the same way as Lemma 2.1.
Lemma 2.3. Let p > 2 be a prime number, h,m ∈ N, m > 1, (h, p) = 1. Then for any
α1, α2 ∈ G the estimate∣∣∣∣∣∣
∑
ξ∈Gpm
e
πi Sp
(α1ξ+α2ξ2+phN(ξ)+p2α3ξ3+...+p2αkξ
k
pm
)∣∣∣∣∣∣ ≤ pm+1 (2.5)
holds.
Proof. Let α1 = a1 + ib1, α2 = a2 + ib2, ξ = x+ iy. Then
Sp(α1ξ) = 2(a1x− b1y), Sp(α2ξ
2) = 2(a2x
2 − a2y2 − 2b2xy).
Hence,
S :=
∑
ξ∈Gpm
e
πiSp
(α1ξ+α2ξ2+phN(ξ)+p2α3ξ3+...
pm
)
=
=
∑
x,y∈Zpm
e
2πi
a1x−b1y+pa2x2−pa2y2−2pb2xy+ph(x2+y2)+p2f(x,y)
pm , (2.6)
where f(x, y) is a polynomial without free term.
We have that (a2 + h, p) = 1 or (a2 − h, p) = 1. Let (a2 + h, p) = 1. We can write
S =
∑
y∈Zpm
e
−2πi b1y−p(h+a
2)y2
pm
∑
x∈Zpm
e
2πi
(a1−2pb2y)x+p(a2h)x2+p2f(x,y)
pm .
It is well-known that the summation on x (or, respectively, y) gives zero if a1 6≡ 0 (mod p) or
b1 6≡ 0 (mod p).
Thus we will set that α1 = p(a1 + ib1). Then we obtain
S =
∑
y∈Zpm
e
−2πi b1y+(h+b2)y2
pm−1
∑
x∈Zpm
e
2πi
(a1−b2y)x+(a2+h)x2+pf(x,y)
pm−1 =
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
GENERALIZED TWISTED KLOOSTERMAN SUM OVER Z[i] 613
= p2
∑
y∈Zpm−1
e
−2πi b1y+(h+b2)y2
pm−1
∑
x∈Zpm−1
e
2πi
(a1−b2y)x+(a2+h)x2+pf(x,y)
pm−1 :=
:=
∑
y∈Zpm−1
e
−2πi b1y+(h+b2)y2
pm−1 · S1(y), (2.7)
say.
The sum S1(y) we can calculate by Lemma 2.1:
S1(y) = ε(m− 1)p
m−1
2 epm−1(f(y0) + p2g(y1) + . . .), (2.8)
where f(y0) ≡
(a1 − 2pb2y)2
2(a2 + h)
≡ a21
2(a2 + h)
− 2phb2
a2 + h
y +
2p2b22
a2 + h
y2 (mod p).
Thereby from (2.7), (2.8) by Lemma 2.1 we infer∣∣∣∣∣∣
∑
ξ∈Gpm
e
πi Sp
(α1ξ+α2ξ2+phN(ξ)+p2α3ξ3+...+p2αkξ
k
pm
)∣∣∣∣∣∣ ≤ pm+1.
Lemma 2.3 is proved.
3. Preliminary result. Let a modulus q1 ∈ Z+, and let χ be a Dirichlet character modulo q1.
Over the ring of Gaussian integers G = Z[i] we define the following generalized twisted Kloosterman
sum with the multiplicative function χ for any q, q ≡ 0 (mod q1):
Kχ(α, β; γ; q) =
∑
x,y∈Gq
χ(N(x))e
2πiSp
(αx+βy
q
)
. (3.1)
Note that χ is not generally a Dirichlet character modulo q, because it can be happened that
χ(N(x)) 6= 0 when (x, q) 6= 1.
In the special case where γ = 1 and q1 = q we obtain the twisted Kloosterman sum with a
character χ defined by
Kχ(α, β; q) =
∑
x,y∈G∗q
xy≡1 (mod q)
χ(x)e
πi Sp
(αx+βy
q
)
. (3.2)
The generalized twisted Kloosterman sum Kχ(α, β; q) has the property of quasimultiplicativity
at q, i.e., for q = q′q′′, (q′, q′′) = 1 the equality
Kχ(α, β; γ; q) = Kχ1(α, β1; γ; q′) ·Kχ2(α, β2; γ; q′′) (3.3)
holds, where χ1, χ2 are characters inducted by character χ, and β1, β2 define from the congruence
β = β1(q
′′)2 + β2(q
′)2 (mod q).
Thus in the sequence it will regard only the Kloosterman sum Kχ(α, β; γ; q) with q = pm, p > 2
be a prime number from Z.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
614 S. VARBANETS
Lemma 3.1. Let p > 2 be a prime number. Suppose q = pm and χ is a Dirichlet character of
conductor pm0 , m0 ≤ m. Then for any Gaussian integers α, β
|Kχ(α, β; pm)| :=
∣∣∣∣∣∣
∑
x,y∈G∗pm
χpm(N(x))epm(<(αx+ βx−1))
∣∣∣∣∣∣ ≤ εpN(p)m/2, (3.4)
where εp =
{
2 if p ≡ 3 (mod 4),
4 if p ≡ 1 (mod 4).
Proof. In the case m = 1 we obtained the required result for principal χ using the Weil’s
result on the estimate of exponential sum on an algebraic curve over finite field, and extended to
nonprincipal character χ (see [13]).
Now we consider the case m > 1. Without loss generality we suppose that (α, β, p) = 1.
Since xy ≡ 1 (mod q), q = pm, we write y = x−1. Then putting x = ξ + pm−1 we have
Kχ(α, β; pm) =
∑
x∈G∗pm
χpm(N(x))epm(<(αx+ βx−1)) =
=
∑
ξ∈G∗
pm−1
∑
η∈Gp
χpm(N(ξ + pm−1η))epm(<(αξ + βξ−1))ep(<((α− βξ−2)η)) =
=
∑
ξ∈G∗
pm−1
χpm(N(ξ))epm(<(αξ + βξ−1))×
×
∑
η∈Gp
χpm(N(1 + pm−1ξ−1η))ep(<((α− βξ−1)η)). (3.5)
Let χpm(A) is defined by the relation
χpm(A) =
e
2πi
ν indA
pm−1(p−1) if (A, p) = 1,
0 if p | A,
where ν ∈ {0, 1, . . . , pm−1(p− 1)− 1}, indA denotes the index of integer A, (A, p) = 1, relatively
to the fixed primitive root modulo pn in Z.
Take into account that ind(1 + p`B) = pm−`−1(p− 1)u−1B with a some u, (u, p) = 1, u is not
depend on B, if (B, p) = 1, and N(1 + pm−1ξ−1η) ≡ 1 + pm−12<(ξ−1η) = 1 + pm−1 Sp(ξ−1η),
we infer
χpm(N(1 + pm−1ξ−1η)) = epm(ν<(ξ−1η)). (3.6)
Now from (3.5), (3.6) it follows
Kχ(α, β; pm) =
∑
ξ∈G∗pm−1
χpm(N(ξ))epm(<(αξ + βξ−1))
∑
η∈Gp
ep
(
<
(
(νξ−1 + α− βξ−2)η
p
))
.
(3.7)
Let Y (α, β, ν) is the set of solutions of the congruence
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
GENERALIZED TWISTED KLOOSTERMAN SUM OVER Z[i] 615
αu2 + νu− β ≡ 0 (mod p), u ∈ G∗pm .
It is clear that |Y (α, β, ν)| ≤
{
2 if p ≡ 3 (mod 4),
4 if p ≡ 1 (mod 4).
From (3.7) we have
Kχ(α, β; pm) = p2
∑
ξ0∈Y
χpm(N(ξ0))×
×
∑
ξ∈Gpm−2
χpm(N(1 + pξ−10 ξ))epm(<(α(ξ0 + pξ) + βξ−10 − βξ
−2
0 pξ + . . .)) =
= p2
∑
ξ0∈Y
χpm(N(ξ0))epm(<(αξ0 + βξ−10 ))×
×
∑
ξ∈Gpm−2
epm−2
(
<
(
(νξ−10 + α− βξ−20 )
p
ξ + βξ−30 ξ2 + pβξ−10 ξ3 + . . .
))
. (3.8)
Now, Lemma 2.2 gives
|Kχ(α, β; pm)| ≤ εpN(p)m/2,
where εp =
{
2 if p ≡ 3 (mod 4),
4 if p ≡ 1 (mod 4).
Lemma 3.1 is proved.
Now, by (3.3) we infer immediately the next corollary.
Corollary 3.1. For any α, β ∈ G and every Dirichlet character χ modulo q, q ∈ N we have
|Kχ(α, β; q)| ≤ τ(q)
√
N((α, β, q))N(q)1/2,
where τ(q) denotes the number of divisors q over G :
τ(q) =
∑
δ|q
∗
1,
(here ∗ denotes that δ runs all nonassociated divisors of q over G).
4. Main results. Now we will investigate the generalized twisted Kloosterman sumKχ(α, β; γ, q)
with parameters α, β, γ ∈ G, χ be a Dirichlet character modulo q1, q1 | q. We put q = q′q′′, where
(q′, q′′) = 1, q′ consists from the same prime numbers as q1, and hence, q1|q′.
From
Gq = Gq′ ×Gq′′
we deduce that χ can considers as a multiplicative function on Gq and it has a canonical factorization
χ = χq′ · χq′′ on Gq, where χq′ is the Dirichlet character modulo q′, viewed as a function on Gq,
and χq′′ is the constant function 1 on Gq.
Thus we have
Kχ(α, β; γ; q′q′′) = Kχq′ (α1, β1; γ; q′) ·Kχq′′ (α2, β2; γ; q′′). (4.1)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
616 S. VARBANETS
Hence, for q =
∏
p|q
pap , q1 =
∏
p|q
pbp we deduce
Kχ(α, β; γ; q) =
∏
p|q′
Kχpa (α, β1p; γ; pap)
∏
p|q′′
Kχ
pb
(α, β2p; γ; pbp), (4.2)
where
β1 ≡ β(q′′)−1 (mod q′), β2 ≡ β(q′)−1 (mod q′′),
β1p ≡ β
(
q
pap
)−1
(mod pap), β2p ≡ β
(
q1
pb
)−1
(mod pb),
moreover in the second product all functions χpb be the constant function 1.
First we consider the multiples of second product.
Let χpb = 1. Write γ = γ1γp, where (γ1, p) = 1, γp = γd11 γ
d2
2 if p ≡ 1 (mod 4), p = p1p2; or
γp = pd if p ≡ 3 (mod 4).
Let p ≡ 3 (mod 4). Using the substitution y = γ1y1 we obtain for d < m
Kχ
pb
(α, β; γ; pm) = Kχ
pb
(α, βγ1; p
d; pm).
The congruence xy ≡ pd (mod pm) has the solutions of type x = pix1, y = pjy1, i + j = d,
x1y1 ≡ 1 (mod pm−d). Thus, grouping the summands of sum in Kχ
pb
(α, βγ1; p
d; pm), according to
i = νp(x) ≤ d, we infer (for χpd being constant function 1)
Kχ
pb
(α, βγ1; p
d; pm) =
d∑
i=0
∑
x∈G∗
pm−d
∑
x1∈G∗
pd−i
∑
y1∈G∗
pi
epm(αpi(x+pm−kx1)+βp
d−i(x−1+pm−ky1)).
Now, the summations over x1, y1 give nonezero only for α ≡ 0 (mod pd−i) and β ≡ 0
(mod pi).
We have therefore obtained the following expression (if χpd = 1):
Kχ
pb
(α, βγ1; γp; p
m) =
N(p)d
∑I2
i=I1
K
(
α
pd−i
,
βγ1
pi
; pm−d
)
if d ≤ νp(α) + νp(β),
0 otherwise,
(4.3)
where I1 = k − νp(α), I2 = νp(β).
For p ≡ 1 (mod 4) we take into account that χpd is the constant function 1 and then
Kχ
pb
(α, βγ1; γp; p
m) = K
(
αpm2 , βγ1p
m
2 ; pd11 ; pm2
)
·K
(
αpm1 , βγ1p
m
1 ; pd22 ; pm2
)
,
where p1p1 ≡ 1 (mod pm2 ), p2p2 ≡ 1 (mod pm1 ).
Thus we have, by the same arguments as for p ≡ 3 (mod 4)
Kχ
pb
(α, βγ1; p
d1
1 pd22 ; pm) =
=
p
d1+d2
∏2
j=1
∑I12
i1=I11
∑I22
i2=I21
Mi1,i2 if dj ≤ νpj (α) + νpj (β), j = 1, 2,
0 otherwise,
(4.4)
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
GENERALIZED TWISTED KLOOSTERMAN SUM OVER Z[i] 617
where
Mk1,k2 = K
(
α
pd1−k11
,
βγ1
pk11
; pm−d11
)
·K
(
α
pd2−k22
,
βγ1
pk22
; pm−d22
)
,
Ij1 = d1 − νpj (α), Ij2 = νpj (β), j = 1, 2.
From (4.3) and Lemma 2.2 we obtain∣∣∣Kχ
pb
(α, β; γ; pm)
∣∣∣ ≤ (d+ 1)(m+ 1)
√
N(αγ, βγ, pm) ·N(pm). (4.5)
For d ≥ m, p ≡ 3 (mod 4) we set
x = pit1, y = pm−it2, t1 ∈ G∗pm−i , t2 ∈ Gpi ,
and get
Kχ(α, βγ1; p
d; pm) =
m∑
i=0
∑
t1∈G∗
pm−i
∑
t2∈Gpi
epm
(
<
(
αpit1 + βγ1p
m−it2
))
. (4.6)
The sum over t2 is N(pi) or 0 according to whether i ≤ νp(β) or not. The sum over t1 is the
Ramanujan sum, so that
∑
t1∈G∗
pm−i
epm(αpit1) =
N(pm−i)−N(pm−i−1) if 0 < m− i ≤ νp(α),
−N(pm−i−1) if m− i = νp(α) + 1,
0 if m− i > νp(α) + 1.
In particular, we see that ith term in (4.6) vanishes unless m− i ≤ νp(α) + 1 and i ≤ νp(β), i.e.,
m− νp(α)− 1 ≤ i ≤ νp(β). Thus the whole expression vanishes unless m ≤ νp(α) + νp(β) + 1.
Thereby in the case d ≥ m
Kχ(α, βγ1; p
d; pm) ≤ (d+ 1) (N (αγ, βγ; pm))1/2N (pm)1/2 . (4.7)
For p ≡ 1 (mod 4) we obtain an analogous result.
Let Kχpa (α, β; γ; pm) be a multiple from the first product in (4.2).
Now, we have χpa(N(x)) = 0 if (x, p) 6= 1, and hence from xy ≡ γ (mod pm), (x, p) = 1,
follows that y ≡ x−1γ (mod p`), and also
Kχpa (α, β; γ; pm) = Kχpa (α, βγ; 1; pm) = Kχpa (α, βγ; pm) (4.8)
holds.
The application of Corollary 3.1 gives
|Kχ(α, β; γ; pm)| ≤ τ(pm)(N(α, βγ; pm))1/2(N(pm))1/2 (4.9)
if χ is a Dirichlet character mod pm0 , 0 < m0 ≤ m.
Multiplying the local estimates (4.7) and (4.8) together, by (4.2) we have
|Kχ(α, β; γ; q)| ≤ τ(γ)τ(q)
√
N(αγ, βγ, q) ·N(p)m/2.
So, we proved the following main theorem.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
618 S. VARBANETS
Theorem 4.1. Let α, β, γ be the Gaussian integers, q > 1 be a positive integer, χ be a Dirichlet
character modulo q1, q1|q. Then the estimate
|Kχ(α, β; γ; q)| ≤ τ(γ)τ(q)
√
N(αγ, βγ, q)q
holds.
Remark 4.1. The method we used to prove the theorem may be applied in the case of q is even.
It is enough to apply the analogues of Lemmas 2.1 and 2.3.
Remark 4.2. In [9] Knightly and Li have shown that the generalized twisted Kloosterman sum
over Z has the estimate
|Kχ(a, b;n; q)| :=
∣∣∣∣∣∣∣∣
∑
x,y∈Zq
xy≡n (mod q)
χ(x)e
2πi
ax+by
q
∣∣∣∣∣∣∣∣ ≤ τ(n)τ(q)(an, bn, q)1/2q1/2q1/2
χ
,
where χ is a Dirichlet character modulo q1 of conductor qχ , q1 | q.
Using the same method of proof as above, we can obtain more precise estimate
|Kχ(a, b;n; q)| ≤ τ(n)τ(q)(an, bn, q)1/2q1/2.
1. Bruggeman R. W., Miatello R. J., Pacharoni I. Estimates for Kloosterman sums for totally real number fields // J.
reine und angew. Math. – 2006. – 535. – S. 103 – 164.
2. Bruggeman R. W. Fourier coefficients of cusp forms // Invent. Math. – 1978. – 445. – P. 1 – 18.
3. Bruggeman R. W., Motohashi Y. Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function
over the Gaussian number field // Funct. Approxim. – 2003. – 31. – P. 23 – 92.
4. Conrad S. On Weil proof of the bound for Kloosterman sums // J. Number Theory. – 2002. – 97, № 2. – P. 439 – 446.
5. Davenport H. On certain exponential sums // J. reine und angew. Math. – 1933. – 169. – S. 158 – 176.
6. Deshouillers I.-M., Iwaniec H. Kloosterman sums and Fourier coefficients of cusp forms // Invent. Math. – 1982. –
70. – P. 219 – 288.
7. Kanemitsu S., Tanigawa Y., Yi. Yuan, Zhang Wenpeng. On general Kloosterman sums // Ann. Univ. sci. budapest. Sec.
comp. – 2003. – 22. – P. 151 – 160.
8. Kloosterman H. D. On the representation of numbers in the form ax2 + by2 + cz2 + dt2 // Acta Math. – 1926. –
49. – P. 407 – 464.
9. Knightlty A., Li C. Kuznetsov’s trace formula and the Hecke eigenvalues of Maas forms // arXiv: 1202. 0189v1[math.
NT], 1 Feb 2012.
10. Kyznetsov N. V. Petterson conjecture for forms of weight zero and the conjecture Linnik. – Khabarovsk, 1977. –
Preprint № 2 (in Russian).
11. Kyznetsov N. V. Petterson conjecture for forms of weight zero and the conjecture Linnik sums of Kloosterman sums
// Math. Sb. – 1980. – 3(153), № 3. – P. 334 – 383 (in Russian).
12. Proskurin N. V. On general Kloosterman sums. – Leningrad, 1980. – Preprint, LOMY, № R-3 (in Russian).
13. Varbanets S. The norm Kloosterman sums over Z[i] // An. Probab. Methods Number Theory. — 2007. — P. 225 – 239.
14. Weil A. On some exponential sums // Proc. Nat. Acad. Sci. USA. – 1948. – 34. – P. 204 – 207.
15. Yi. Yuan, Zhang Wenpeng. On the generalization of a problem of D. H. Lehmer // Kyushu J. Math. – 2002. – 56. –
P. 235 – 241.
Received 13.11.12,
after revision — 22.10.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 5
|