On the Gauss sums and generalized Bernoulli numbers
Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508552044478464 |
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| author | Gao, J. Liu, He-guo Гао, Дж. Лю, Хе-го |
| author_facet | Gao, J. Liu, He-guo Гао, Дж. Лю, Хе-го |
| author_sort | Gao, J. |
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| datestamp_date | 2020-03-18T19:30:57Z |
| description | Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas. |
| first_indexed | 2026-03-24T02:27:01Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
H. Liu (Northwest Univ., China),
J. Gao (Xi’an Jiaotong Univ., China)
ON THE GAUSS SUMS AND GENERALIZED BERNOULLI NUMBERS*
ПРО СУМИ ГАУССА ТА УЗАГАЛЬНЕНI ЧИСЛА БЕРНУЛЛI
Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of
Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas.
Iз використанням примiтивних характерiв, сум Гаусса та суми Рамануджана вивчено два гiбридних середнiх значення
сум Гаусса й узагальнених чисел Бернуллi та отримано двi асимптотичнi формули.
1. Introduction. Let χ be a Dirichlet character modulo q ≥ 3. For any integer n, the Gauss sum
G(n, χ) is defined as following:
G(n, χ) =
q∑
a=1
χ(a)e
(
an
q
)
,
where e(y) = e2πiy. Especially for n = 1, we write τ(χ) =
∑q
a=1
χ(a)e
(
a
q
)
. The various
properties and applications of τ(χ) appear in many analytic number theory books (see reference [1]).
Maybe the most important property of τ(χ) is that if χ is a primitive character modulo q, then
|τ(χ)| = √q. If χ is a non-primitive character modulo q, τ(χ) also appears many good value
distribution properties in some problems of weighted mean value. For example, Y. Yi and W. Zhang
[2] studied the 2k-th power mean of inversion of L-functions with the weight of Gauss sums, and
gave some interesting formulae.
Let χ be a non-principal Dirichlet character modulo q. The generalized Bernoulli numbers Bn,χ
is defined by the following:
q∑
a=1
χ(a)
teat
eqt − 1
=
∞∑
n=0
Bn,χ
n!
tn.
This sequence of numbers has considerable fascination and importance. The definition and basic
properties of generalized Bernoulli numbers can be found in [3]. H. Liu and W. Zhang [4] used the
properties of primitive characters and the mean value theorems of Dirichlet L-functions to study the
hybrid mean value ∑
χ 6=χ0
χ mod q
τm (χ)Bm
n,χ,
*Supported by the National Natural Science Foundation of China under Grant No.10901128, the Specialized Research
Fund for the Doctoral Program of Higher Education of China under Grant No. 20090201120061, the Natural Science
Foundation of the Education Department of Shaanxi Province of China under Grant No. 09JK762, and the Fundamental
Research Funds for the Central University.
c© H. LIU, J. GAO, 2012
848 ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON THE GAUSS SUMS AND GENERALIZED BERNOULLI NUMBERS 849
and give a sharper asymptotic formula.
It might be interesting to study more mean value of the Gauss sums and generalized Bernoulli
numbers. In this paper, we use the properties of primitive characters, Gauss sums and Ramanujan
sum to study two hybrid mean value of Gauss sums and generalized Bernoulli numbers, and give
two interesting formulae. That is, we shall prove the following.
Theorem 1. Let q ≥ 3 be an integer. Then for any given positive integers n > 1 and m we
have ∑
χ 6=χ0
τ(χ)6=0
Bm
n,χ
τm(χ)
=
(−1)m2m−1(n!)m
(2πi)nm
qm(n−1)−1φ2(q)
∏
p‖q
(
1 +
1
p− 1
)
+O
(
qm(n−1)d(q)
)
,
where
∑
χ6=χ0
τ(χ)6=0
denotes the summation over all non-principal characters modulo q with τ(χ) 6= 0,∏
p‖q
denotes the product over all prime divisors p of q with p | q and p2 - q, φ(q) is the Euler
function, d(q) denotes the divisor function, and the O-constant depends on m and n.
Theorem 2. For any fixed positive integers m > 2 and n > 1, we have
∑
χ1 mod q
∑
χ2 6=χ0
τ(χ1χ2)Bn,χ2
m =
(−1)(n+1)m(n!)m
(2πi)nm
qm(n−1)φ2m(q)+
+O
(
qm(n+1)−1d(q)
)
+O
(
qm(n+1/2)+1dm(q)
)
,
where the O-constant depends on m and n.
2. Some lemmas. To complete the proof of the theorems, we need the following lemmas.
Lemma 1. For any integer q ≥ 3, let χ be a non-primitive character modulo q, and q∗ denote
the conductor of χ with χ⇐⇒ χ∗. If (n, q) > 1, we have
G(n, χ) =
χ∗
(
n
(n, q)
)
χ∗
(
q
q∗(n, q)
)
µ
(
q
q∗(n, q)
)
φ(q)φ−1
(
q
(n, q)
)
τ(χ∗), q∗ =
q1
(n, q1)
,
0, q∗ 6= q1
(n, q1)
,
where µ(n) is the Möbius function, and q1 is the largest divisor of q that has the same prime factors
with q∗.
If (n, q) = 1, then we have
G(n, χ) = χ∗(n)χ∗
(
q
q∗
)
µ
(
q
q∗
)
τ(χ∗).
Proof. See reference [5].
Lemma 2. Let q and r be integers with q ≥ 3 and (r, q) = 1, χ be a Dirichlet character
modulo q. Then we have the identities∑∗
χ mod q
χ(r) =
∑
d|(q,r−1)
µ
(q
d
)
φ(d)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
850 H. LIU, J. GAO
and
J(q) =
∑
d|q
µ(d)φ
(q
d
)
,
where
∑∗
χ mod q
denotes the summation over all primitive characters modulo q, and J(q) denotes
the number of primitive characters modulo q.
Proof. This is Lemma 3 of [6]. Also one can see Lemma 4 of [7].
Lemma 3. Let q = uv, where (u, v) = 1, u be a square-full number or u = 1, v be a
square-free number. Then for any given positive integers n > 1 and m we have
∑
d|v
∑∗
χ mod ud
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=−∞
r 6=0
χ(r)
rn
m
=
2m−1φ2(q)
q
∏
p‖q
(
1 +
1
p− 1
)
+O (d(q)).
Proof. It is easy to show that
∑
d|v
∑∗
χ mod ud
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=−∞
r 6=0
χ(r)
rn
m
=
=
∑
d|v
∑∗
χ mod ud
[1 + χ(−1)(−1)n]m
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=1
χ(r)
rn
m =
=
2m
∑
d|v
∑∗
χ mod ud
χ(−1)=1
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=1
χ(r)
rn
m , if 2 | n,
2m
∑
d|v
∑∗
χ mod ud
χ(−1)=−1
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=1
χ(r)
rn
m , if 2 - n.
Let τm(r) denote the m-th divisor function (i.e., the number of positive integer solutions of the
equation r = r1r2 . . . rm). Note that J(u) = φ2(u)/u, if u is a square-full number. Then using the
methods of Lemma 3 in [4] and Lemma 2 in this paper we have
∑
d|v
∑∗
χ mod ud
χ(−1)=−1
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=1
χ(r)
rn
m =
=
∑
d|v
∑
t1| vd
. . .
∑
tm| vd
µ(t1) . . . µ(tm)φ(t1) . . . φ(tm)
tn1 . . . t
n
m
+∞∑
r=1
τm(r)
rn
∑∗
χ mod ud
χ(−1)=−1
χ(t1 . . . tm)χ(r) =
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON THE GAUSS SUMS AND GENERALIZED BERNOULLI NUMBERS 851
=
1
2
∑
d|v
∑
s|ud
µ
(
ud
s
)
φ(s)
∑
t1| vd
. . .
∑
tm| vd
+∞∑
r=1
t1...tmr≡1 mod s
µ(t1) . . . µ(tm)φ(t1) . . . φ(tm)τm(r)
(t1 . . . tmr)
n −
−1
2
∑
d|v
∑
s|ud
µ
(
ud
s
)
φ(s)
∑
t1| vd
. . .
∑
tm| vd
+∞∑
r=1
t1...tmr≡−1 mod s
µ(t1) . . . µ(tm)φ(t1) . . . φ(tm)τm(r)
(t1 . . . tmr)
n =
=
1
2
∑
d|v
J(ud) +O
∑
d|v
∑
s|ud
φ(s)
+∞∑
l=1
1
(ls+ 1)n−1+ε
+
+O
∑
d|v
∑
s|ud
φ(s)
+∞∑
l=1
1
(ls− 1)n−1+ε
=
=
1
2
∑
d|v
J(ud) +O (d(q)) =
φ2(u)
2u
∑
d|v
J(d) +O (d(q)) =
=
φ2(u)
2u
∏
p|v
(p− 1) +O (d(q)) =
φ2(u)
2u
∏
p|v
[
(p− 1)2
p
(
1 +
1
p− 1
)]
+O (d(q)) =
=
φ2(q)
2q
∏
p‖q
(
1 +
1
p− 1
)
+O (d(q)) .
Similarly we can get
∑
d|v
∑∗
χ mod ud
χ(−1)=1
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=1
χ(r)
rn
m =
φ2(q)
2q
∏
p‖q
(
1 +
1
p− 1
)
+O (d(q)) .
So we have
∑
d|v
∑∗
χ mod ud
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=−∞
r 6=0
χ(r)
rn
m
=
2m−1φ2(q)
q
∏
p‖q
(
1 +
1
p− 1
)
+O (d(q)) .
Lemma 3 is proved.
Lemma 4. Let χ be a Dirichlet character modulo q and n > 1 be a fixed integer. Then we
have
+∞∑
r=−∞
r 6=1
G(r, χ)
(r − 1)n
=
(−1)nφ(q) +O (d(q)) , if χ = χ0 is the principal character,
O
(
q1/2d(q)
)
, otherwise.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
852 H. LIU, J. GAO
Proof. First we suppose that χ is a non-principal character modulo q. Noting that
G(r, χ) ≤ (r, q)q1/2,
then we have
+∞∑
r=−∞
r 6=1
G(r, χ)
(r − 1)n
� q1/2
+∞∑
r=2
(r, q)
(r − 1)n
= q1/2
∑
d|q
∑
2/d≤l<+∞
d
(ld− 1)n
=
= q1/2
∑
d|q
1
dn−1
∑
2/d≤l<+∞
1
(l − 1/d)n
� q1/2d(q).
If χ = χ0 is the principal character modulo q, then G(r, χ0) = Cq(r) is the Ramanujan sum.
Noting that
Cq(r) =
∑
d|(q,r)
dµ
(q
d
)
,
then we have
+∞∑
r=−∞
r 6=1
G(r, χ0)
(r − 1)n
=
+∞∑
r=−∞
r 6=1
1
(r − 1)n
∑
d|(q,r)
dµ
(q
d
)
=
∑
d|q
+∞∑
l=−∞
ld6=1
dµ (q/d)
(ld− 1)n
=
= (−1)n
∑
d|q
dµ
(q
d
)
+
∑
d|q
+∞∑
l=−∞
ld6=1
l 6=0
dµ (q/d)
(ld− 1)n
=
= (−1)n
∑
d|q
dµ
(q
d
)
+O (d(q)) = (−1)nφ(q) +O (d(q)) .
Lemma 4 is proved.
3. Proof of the theorems. In this section, we complete the proof of the theorems. Let q ≥ 3
be an integer, and χ be a Dirichlet character modulo q. The generalized Bernoulli numbers can be
expressed in terms of Bernoulli polynomials as
Bn,χ = qn−1
q∑
a=1
χ(a)Bn
(
a
q
)
.
From Theorem 12.19 of [1] we also have
Bn(x) = −
n!
(2πi)n
+∞∑
r=−∞
r 6=0
e (rx)
rn
, if 0 < x ≤ 1.
Therefore
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON THE GAUSS SUMS AND GENERALIZED BERNOULLI NUMBERS 853
Bn,χ = qn−1
q∑
a=1
χ(a)
− n!
(2πi)n
+∞∑
r=−∞
r 6=0
e
(
ar
q
)
rn
= −n!q
n−1
(2πi)n
+∞∑
r=−∞
r 6=0
G(r, χ)
rn
. (1)
Let q = uv, where (u, v) = 1, u be a square-full number or u = 1, v be a square-free number.
Let q∗ denote the conductor of χ with χ⇐⇒ χ∗, then
τ(χ) = χ∗
(
q
q∗
)
µ
(
q
q∗
)
τ(χ∗) 6= 0
if and only if q∗ = ud, where d | v. So from Lemmas 1 and 3 we have
∑
χ 6=χ0
τ(χ)6=0
Bm
n,χ
τm(χ)
=
∑
d|v
∑∗
χ mod ud
−n!qn−1(2πi)n
∑
t| v
d
χ
( v
dt
)
µ
( v
dt
)
φ(q)τ(χ)
tnφ
(q
t
) +∞∑
r=−∞
r 6=0
χ(r)
rn
m
χm
(v
d
)
µm
(v
d
)
τm(χ)
.
Noting that
χ
(v
d
)
= χ
( v
dt
)
χ(t), µ
(v
d
)
= µ
( v
dt
)
µ(t),
then we get
χ
( v
dt
)
µ
( v
dt
)
φ(q)
tnφ
(q
t
)
χ
(v
d
)
µ
(v
d
) =
φ(t)
tnχ(t)µ(t)
=
χ(t)µ(t)φ(t)
tn
.
Therefore by Lemma 3 and the above we have
∑
χ 6=χ0
τ(χ)6=0
Bm
n,χ
τm(χ)
=
(−1)m(n!)mqm(n−1)
(2πi)nm
∑
d|v
∑∗
χ mod ud
∑
t| v
d
χ(t)µ(t)φ(t)
tn
+∞∑
r=−∞
r 6=0
χ(r)
rn
m
=
=
(−1)m2m−1(n!)m
(2πi)nm
qm(n−1)−1φ2(q)
∏
p‖q
(
1 +
1
p− 1
)
+O
(
qm(n−1)d(q)
)
.
Theorem 1 is proved.
From the orthogonality relations for character sums, formula (1) and Lemma 4 we can get
∑
χ2 6=χ0
τ(χ1χ2)Bn,χ2 = −n!q
n−1
(2πi)n
+∞∑
r=−∞
r 6=0
1
rn
∑
χ2 6=χ0
τ(χ1χ2)G(r, χ2) =
= −n!q
n−1
(2πi)n
+∞∑
r=−∞
r 6=0
1
rn
q∑
a=1
χ1(a)e
(
a
q
) q∑
b=1
e
(
br
q
) ∑
χ2 6=χ0
χ2(a)χ2(b) =
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
854 H. LIU, J. GAO
= −n!q
n−1φ(q)
(2πi)n
+∞∑
r=−∞
r 6=0
1
rn
q∑
a=1
χ1(a)e
(
a(1 + r)
q
)
+O (qn) =
= −n!q
n−1φ(q)
(2πi)n
+∞∑
r=−∞
r 6=1
G(r, χ1)
(r − 1)n
+O (qn) =
=
(−1)n+1n!qn−1φ2(q)
(2πi)n
+O (qnd(q)) , if χ1 = χ0 is the principal character;
O
(
qn+1/2d(q)
)
, otherwise.
Then we have
∑
χ1 mod q
∑
χ2 6=χ0
τ(χ1χ2)Bn,χ2
m =
=
∑
χ2 6=χ0
τ(χ0χ2)Bn,χ2
m +
∑
χ1 6=χ0
∑
χ2 6=χ0
τ(χ1χ2)Bn,χ2
m =
=
(−1)(n+1)m(n!)m
(2πi)nm
qm(n−1)φ2m(q) +O
(
qm(n+1)−1d(q)
)
+O
(
qm(n+1/2)+1dm(q)
)
,
which is valid for m > 2.
Theorem 2 is proved.
1. Apostol T. M. Introduction to analytic number theory. – New York: Springer, 1976.
2. Yi Y., Zhang W. On the 2k-th power mean of inversion of L-functions with the weight of Gauss sums // Acta Math.
Sinica (English Ser.). – 2004. – 20. – P. 175 – 180.
3. Leopoldt A. W. Eine Verallgemeinerung der Bernoullischen Zahlen // Abhand. Math. Semin. Univ. Hamburg. – 1958.
– 22. – S. 131 – 140.
4. Liu H., Zhang W. On the hybrid mean value of Gauss sums and generalized Bernoulli numbers // Proc. Jap. Acad.
Ser. A. Math. Sci. – 2004. – 80. – P. 113 – 115.
5. Pan C., Pan C. Goldbach conjecture. – Beijing: Sci. Press, 1981.
6. Zhang W. On a Cochrane sum and its hybrid mean value formula // J. Math. Anal. and Appl. – 2002. – 267. –
P. 89 – 96.
7. Zhang W., Liu H. A note on the Cochrane sum and its hybrid mean value formula // J. Math. Anal. and Appl. – 2003.
– 288. – P. 646 – 659.
Received 27.09.11
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
|
| id | umjimathkievua-article-2622 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:27:01Z |
| publishDate | 2012 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/00/6099ab9320090567d1e0f36896260e00.pdf |
| spelling | umjimathkievua-article-26222020-03-18T19:30:57Z On the Gauss sums and generalized Bernoulli numbers Про суми Гаусса та узагальненi числа Бернуллi Gao, J. Liu, He-guo Гао, Дж. Лю, Хе-го Using the properties of primitive characters, Gauss sums, and the Ramanujan sum, we study two hybrid mean values of Gauss sums and generalized Bernoulli numbers and give two asymptotic formulas. Iз використанням примiтивних характерiв, сум Гаусса та суми Рамануджана вивчено два гiбридних середнiх значення сум Гаусса й узагальнених чисел Бернуллi та отримано двi асимптотичнi формули. Institute of Mathematics, NAS of Ukraine 2012-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2622 Ukrains’kyi Matematychnyi Zhurnal; Vol. 64 No. 6 (2012); 848-854 Український математичний журнал; Том 64 № 6 (2012); 848-854 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2622/2003 https://umj.imath.kiev.ua/index.php/umj/article/view/2622/2004 Copyright (c) 2012 Gao J.; Liu He-guo |
| spellingShingle | Gao, J. Liu, He-guo Гао, Дж. Лю, Хе-го On the Gauss sums and generalized Bernoulli numbers |
| title | On the Gauss sums and generalized Bernoulli numbers |
| title_alt | Про суми Гаусса та узагальненi числа Бернуллi |
| title_full | On the Gauss sums and generalized Bernoulli numbers |
| title_fullStr | On the Gauss sums and generalized Bernoulli numbers |
| title_full_unstemmed | On the Gauss sums and generalized Bernoulli numbers |
| title_short | On the Gauss sums and generalized Bernoulli numbers |
| title_sort | on the gauss sums and generalized bernoulli numbers |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2622 |
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