Journal of Inequalities in Pure and
Applied Mathematics
AN INEQUALITY FOR THE CLASS NUMBER
OLIVIER BORDELLÈS
2 allée de la combe, la Boriette
43000 AIGUILHE
FRANCE
volume 7, issue 3, article 87,
2006.
Received 05 October, 2005;
accepted 10 March, 2006.
Communicated by: J. Sándor
EMail: borde43@wanadoo.fr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
307-05
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Abstract
We prove in an elementary way a new inequality for the average order of the
Piltz divisor function with application to class number of number fields.
2000 Mathematics Subject Classification: 11N99, 11R29.
Key words: Piltz divisor function, Class number.
I would like to thank my wife Véronique for her help.
An Inequality for the Class
Number
Olivier Bordellès
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
The Case n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4
Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References
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1.
Introduction
It could be interesting to use tools from analytic number theory to solve problems of algebraic number theory. For example, let K be a number field of degree
n, signature (r1 , r2 ), class number hK , regulator RK , and wK is the number of
1/2
roots of unity in K, ζK the Dedekind zeta function, AK := 2−r2 π −n/2 dK where
dK is the absolute value of the discriminant of K. The following formula, valid
for any real number σ > 1,
σ r1 σ
(1.1) AK Γ
Γr2 (σ) ζK (σ)
2
n
o
XZ
1−σ
dy
2r1 hK RK
σ/2
2
kyk + kyk
=
+
e−g(a,y) ,
σ (σ − 1) wK a6=0 kyk≥1
y
An Inequality for the Class
Number
Olivier Bordellès
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where g (a, y) is a certain function depending on a nonzero integral ideal a and
vector y := (y1 , . . . , yr1 +r2 ) ∈ (R+ )r1 +r2 (here kyk := max |yi |), is the generalization of the well-known formula
∞ Z ∞n
σ o
X
1−σ
1
2 dy
−σ/2
ζ (σ) =
+
π
Γ
y σ/2 + y 2 e−πn y
2
σ (σ − 1) n=1 1
y
for the classical Riemann zeta function. Since the integrand in (1.1) is positive,
we get
σ (1.2)
hK RK ≤ σ (σ − 1) wK 2−r1 AσK Γr1
Γr2 (σ) ζK (σ)
2
for any real number σ > 1. The study of the function on the right-hand side of
(1.2) provides upper bounds for hK RK (see [3] for example) .
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In a more elementary way, one can connect the class number hK with the
Piltz divisor function τn by using the following result ([1]):
Lemma 1.1. Let bK > 0 be a real number such that every class of ideals of
K contains a nonzero integral ideal with norm ≤ bK . If τn is the Piltz divisor
function, then:
X
τn (m) .
hK ≤
m≤bK
P Recall that τn is defined by the relations τ1 (m) = m and τn (m) =
d|m τn−1 (d) (n ≥ 2). This function has been studied by many authors (see [6]
for a good survey of its properties). A standard argument from analytic number
theory gives if n ≥ 4
n−1 X
τn (m) = xPn−1 (log x) + Oε x n+2 +ε ,
m≤x
An Inequality for the Class
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1
where Pn−1 is a polynomial of degree n − 1 and leading coefficient (n−1)!
. For
some improvements of the error term and related results, see [4]. Note that the
Lindelöf Hypothesis is equivalent to αn = (n − 1) / (2n) for any n = 2, 3, . . .
where αn is the least number such that
X
τn (m) − xPn−1 (log x) = Oε xαn +ε .
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m≤x
If we are interested in finding upper bounds of the form
X
τn (m) n x (log x)n−1 ,
m≤x
one mostly uses arguments based upon induction and the following inequality:
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Lemma 1.2. We set Sn (x) :=
P
m≤x τn
(m) . Then:
Z x
Sn+1 (x) ≤ Sn (x) + x
t−2 Sn (t) dt.
1
Proof. It suffices to use the definition above, interchange the summations and
integrate by parts.
Using this lemma, it is easy to show by induction the following bound:
X
x
τn (m) ≤
(log x + n − 1)n−1
(n − 1)!
m≤x
which enables us to obtain Lenstra’s bound again (see [2]), namely:
(1.3)
hK ≤
bK
(log bK + n − 1)n−1 .
(n − 1)!
In what follows, n is a positive integer and we set
X
Sn (x) :=
τn (m)
m≤x
for any real number x ≥ 1. bK is a positive real number always satisfying the
hypothesis of Lemma 1.1. K is a number field of degree n and class number
hK . dK is the absolute value of the discriminant of K. For some tables giving
values of bK , see [7]. The functions ψ and ψ2 are defined by
1
ψ (t) = t − [t] − ,
2
Z t
1
ψ 2 (t)
,
ψ2 (t) =
ψ (u) du + =
8
2
0
An Inequality for the Class
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Olivier Bordellès
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where [t] denotes the integral part of t. Recall that we have for all real numbers
t:
1
|ψ (t)| ≤ ,
2
1
0 ≤ ψ2 (t) ≤ .
8
We denote by γ and γ1 the Euler-Mascheroni constant and the first Stieltjes
constant, defined respectively by:
!
n
X
1
γ = lim
− log n ,
n→∞
k
k=1
!
n
X
log k (log n)2
γ1 = lim
−
.
n→∞
k
2
k=1
The following results are well-known (see [5] for example):
0.577215 < γ < 0.577216,
−0.072816 < γ1 < −0.072815,
1
γ = −2
2
Z
1
∞
ψ2 (t)
dt
t3
and
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Z
(1.5)
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and
(1.4)
An Inequality for the Class
Number
γ1 = −
1
∞
2 log t − 3
ψ2 (t) dt.
t3
J. Ineq. Pure and Appl. Math. 7(3) Art. 87, 2006
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2.
Results
Theorem 2.1. Let n ≥ 3 be an integer. For any real number x ≥ 13, we have:
X
x
τn (m) ≤
(log x + n − 2)n−1 .
(n
−
1)!
m≤x
Applying this result with Lemma 1.1 allows us to improve upon (1.3) :
Theorem 2.2. Let K be a number field of degree n ≥ 3. If bK ≥ 13 satisfies the
hypothesis of Lemma 1.1, then:
bK
hK ≤
(log bK + n − 2)n−1 .
(n − 1)!
An Inequality for the Class
Number
Olivier Bordellès
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3.
The Case n = 3
The aim of this section is to show that the result of Theorem 2.1 is true for
n = 3. Hence we will prove the following inequality for S3 :
Lemma 3.1. For any real number x ≥ 13, we have:
S3 (x) ≤
x
(log x + 1)2 .
2
We first check this result for 13 ≤ x ≤ 670 with the PARI/GP system [8],
and then suppose x > 670. The lemma will be a direct consequence of the
following estimation:
Lemma 3.2. For any real number x > 670, we have:
)
(
(log x)2
S3 (x) = x
+ (3γ − 1) log x + 3γ 2 − 3γ − 3γ1 + 1 + R (x)
2
where:
|R (x)| ≤ 2.36 x2/3 log x.
An Inequality for the Class
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Olivier Bordellès
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The proof of this lemma needs some technical results:
Lemma 3.3. Let x, y ≥ 1 be real numbers.
3/2
(i) If e
≤ y ≤ x, then we have:
X1
k≤y
k
log
x
k
= log x log y −
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(log y)2
+ γ log x − γ1 + R1 (x, y)
2
J. Ineq. Pure and Appl. Math. 7(3) Art. 87, 2006
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with:
|R1 (x, y)| ≤
log (x/y) log x
+
.
2y
4y 2
(ii)
S2 (y) = y log y + (2γ − 1) y + R2 (y)
with:
1
|R2 (y)| ≤ y 1/2 + .
2
(iii)
X τ (n)
n≤y
n
(log y)2
=
+ 2γ log y + γ 2 − 2γ1 + R3 (y)
2
|R3 (y)| ≤
1
y 1/2
1
+ .
y
Proof. (i) By the Euler-MacLaurin summation formula, we get:
k≤y
k
log
Olivier Bordellès
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with:
X1
An Inequality for the Class
Number
x
k
Z
x
y
log x
1
ψ (y)
x
=
+
log
dt −
log
2
t
y
y
1 t
Z y
ψ2 (y)
x
2 log (x/t) + 3
−
log
+1 −
ψ2 (t) dt
2
y
y
t3
1
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Z ∞
(log y)2
1
ψ2 (t)
= log x log y −
+
−2
dt log x
2
2
t3
1
Z ∞
2 log t − 3
ψ (y)
x
ψ2 (y)
x
+
ψ2 (t) dt −
log
log
−
+1
t3
y
y
y2
y
1
Z ∞
Z ∞
ψ2 (t)
2 log t − 3
+ 2 log x
dt −
ψ2 (t) dt
3
t
t3
y
y
and using (1.4) and (1.5) we get:
X1
k≤y
k
log
x
k
2
= log x log y −
(log y)
+ γ log x − γ1 + R1 (x, y)
2
and since e3/2 ≤ y ≤ x, we have:
log (x/y) log (x/y) + 1 log x log y − 1
|R1 (x, y)| ≤
+
+
+
2y
8y 2
8y 2
8y 2
log (x/y) log x
=
+
.
2y
4y 2
(ii) This result is well-known (see [1] for example).
(iii) Using a result from [5], we have for any real number y ≥ 1 :
3
1
y −3/2 y −2
1
1
−1/2
−1
−1/2
−y
−
+
y −
−
≤ R3 (y) ≤ y
+
+
y −1
4 8e3
8
64
2 8e3
which concludes the proof of Lemma 3.3.
An Inequality for the Class
Number
Olivier Bordellès
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Proof of Lemmas 3.1 and 3.2. The Dirichlet hyperbola principle and the estimations of Lemma 3.3 give, for any real number e3/2 ≤ T < x:
hxi
x
X x
X
S3 (x) =
S2
τ (n)
+
− [T ] S2
n
n
T
n≤T
n≤x/T
x
X x
X τ (n) 1 x x
=
log
+ (2γ − 1) + R4 (x, n) + x
− S2
n
n
n
n
2
T
n≤T
n≤x/T
x 1 x
x
x
X
−
τ (n) ψ
− T S2
+ S2
+ ψ (T ) S2
n
T
2
T
T
n≤x/T
X x
x
x
=
log
+ (2γ − 1) + R4 (x, n)
n
n
n
n≤T
x
X τ (n)
+x
− T S2
+ R5 (x, T )
n
T
n≤x/T
with
r
x 1
+
n 2
r
x
x
x
x
x 1
|R5 (x, T )| ≤ S2
≤ log
+ (2γ − 1) +
+
T
T
T
T
T
2
|R4 (x, n)| ≤
and hence:
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Olivier Bordellès
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(
(log T )2
S3 (x) = x log x log T −
+ γ log x
2
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− γ1 + R6 (x, T ) + (2γ − 1) (log T + γ + R7 (T ))
o
X
+
R4 (x, n)
n≤T
(
+x
)
(log (x/T ))
+ 2γ log
+ γ 2 − 2γ1 + R8 (x, T )
2
T
x
+ R5 (x, T ) − x log
− (2γ − 1) x − T R9 (x, T )
T
2
x
with, if e3/2 ≤ T < x :
log (x/T ) log x
+
2T
4T 2
1
|R7 (T )| ≤
T
r
T
T
|R8 (x, T )| ≤
+
x
rx
x 1
|R9 (x, T )| ≤
+
T
2
|R6 (x, T )| ≤
Olivier Bordellès
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and thus:
(
S3 (x) = x
An Inequality for the Class
Number
)
(log x)2
+ (3γ − 1) log x + 3γ 2 − 3γ − 3γ1 + 1
2
+ xR6 (x, T ) + (2γ − 1) xR7 (T ) + R10 (x, T )
+ xR8 (x, T ) + R5 (x, T ) − T R9 (x, T )
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with
|R10 (x, T )| ≤
X
|R4 (x, n)|
n≤T
√ X 1
T
√ +
x
2
n
n≤T
√
√
T
≤ 2 xT − x +
2
≤
and therefore:
(
S3 (x) = x
An Inequality for the Class
Number
)
(log x)2
2
+ (3γ − 1) log x + 3γ − 3γ − 3γ1 + 1 + R11 (x, T )
2
Olivier Bordellès
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with:
Contents
√
√
x log (x/T ) x log x
+
+
4
xT − x
|R11 (x, T )| ≤
2
2T
4T
r
x
x
x
1
2x
log
+ 2 (2γ − 1) +
+ 2T + .
+
T
T
T
T
2
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We choose:
T =x
1/3
,
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which gives:
Page 13 of 17
(
S3 (x) = x
JJ
J
)
(log x)2
+ (3γ − 1) log x + 3γ 2 − 3γ − 3γ1 + 1
2
+ R12 (x) ,
J. Ineq. Pure and Appl. Math. 7(3) Art. 87, 2006
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where:
5
1
1
|R12 (x)| ≤ x2/3 log x + 2 (2γ + 1) x2/3 − x1/2 + x1/3 log x + 3x1/3 +
3
4
2
2/3
≤ 2.36 x log x
since x > 670. This concludes the proof of Lemma 3.2, and then of Lemma
3.1.
An Inequality for the Class
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Olivier Bordellès
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4.
Proof of Theorem 2.1
We first need the following simple bounds:
Lemma 4.1. For any integer n ≥ 3, we have:
n
Z 13
n3
1
1
−2
t Sn (t) dt <
.
≤
n+
4
n!
2
1
Proof. This follows from straightforward computations which give:
Z 13
7 3 2281 2 90283
1
t−2 Sn (t) dt =
n +
n +
n+1−
624
9360
90090
13
1
3
n
<
4
since n ≥ 3. The second inequality follows from studying the sequence (un )
defined by
n3 × n!
.
un =
4 (n + 1/2)n
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Olivier Bordellès
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We get:
Close
n
un+1
2 (n + 1)4 2n + 1
= 3
un
n (2n + 3) 2n + 3
n
512
2
512e−1
≤
1−
≤
<1
243
2n + 3
243
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and hence (un ) is decreasing, and thus:
un ≤ u3 =
324
≤ 1,
343
which concludes the proof of Lemma 4.1.
Proof of Theorem 2.1. We use induction, the result being true for n = 3 by
Lemma 3.1. Now suppose the inequality is true for some integer n ≥ 3. By
Lemmas 1.2, 4.1 and the induction hypothesis, we get:
Sn+1 (x)
Z
13
−2
Z
An Inequality for the Class
Number
Olivier Bordellès
x
−2
≤ Sn (x) + x
t Sn (t) dt + x
t Sn (t) dt
1
13
)
(
n
Z x
1
1
1
(log t + n − 2)n−1
(log x + n − 2)n−1
+
n+
+
dt
≤x
(n − 1)!
n!
2
(n − 1)! 13
t
(
(log x + n − 2)n (log x + n − 2)n−1
+
=x
n!
(n − 1)!
n
1
1
−2 n
+
n+
− n + log 13e
n!
2
x ≤
(log x + n − 2)n + (n − 1) (log x + n − 2)n−1
n!
x
≤
(log x + n − 1)n .
n!
The proof of Theorem 2.1 is now complete.
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References
[1] O. BORDELLÈS, Explicit upper bounds for the average order of dn (m)
and application to class number, J. Inequal. Pure and Appl. Math., 3(3)
(2002), Art. 38. [ONLINE: http://jipam.vu.edu.au/article.
php?sid=190]
[2] H.W. LENSTRA Jr., Algorithms in algebraic number theory, Bull. Amer.
Math. Soc., 2 (1992), 211–244.
[3] S. LOUBOUTIN, Explicit bounds for residues of Dedekind zeta functions,
values of L−functions at s = 1, and relative class number, J. Number Theory, 85 (2000), 263–282.
[4] D.S. MITRINOVIĆ, J. SÁNDOR AND B. CRSTICI, Handbook of Number
Theory I, Springer-Verlag, 2nd printing, (2005).
[5] H. RIESEL AND R.C. VAUGHAN, On sums of primes, Arkiv för Mathematik, 21 (1983), 45–74.
[6] J. SÁNDOR, On the arithmetical function dk (n), L’Analyse Numér. Th.
Approx., 18 (1989), 89–94.
[7] R. ZIMMERT, Ideale kleiner norm in Idealklassen und eine Regulatorabschätzung, Fakultät für Mathematik der Universität Bielefield, Dissertation (1978).
[8] PARI/GP, Available by anonymous ftp from: ftp://megrez.math.
u-bordeaux.fr/pub/pari.
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Olivier Bordellès
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