Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 7, Issue 3, Article 87, 2006
AN INEQUALITY FOR THE CLASS NUMBER
OLIVIER BORDELLÈS
2 ALLÉE DE LA COMBE , LA B ORIETTE
43000 AIGUILHE
FRANCE
borde43@wanadoo.fr
Received 05 October, 2005; accepted 10 March, 2006
Communicated by J. Sándor
A BSTRACT. 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.
Key words and phrases: Piltz divisor function, Class number.
2000 Mathematics Subject Classification. 11N99, 11R29.
1. I NTRODUCTION
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 roots of unity in K, ζK the Dedekind zeta
1/2
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 (1.1)
AσK Γr1
Γr2 (σ) ζK (σ)
2
n
o
XZ
1−σ
2r1 hK RK
dy
=
+
kykσ/2 + kyk 2 e−g(a,y) ,
σ (σ − 1) wK a6=0 kyk≥1
y
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
σ/2
2
π
Γ
ζ (σ) =
+
y +y
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
ISSN (electronic): 1443-5756
c 2006 Victoria University. All rights reserved.
I would like to thank my wife Véronique for her help.
307-05
2
O LIVIER B ORDELLÈS
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) .
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
hK ≤
τn (m) .
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
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 .
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:
P
Lemma 1.2. We set Sn (x) := 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
(log x + n − 1)n−1
τn (m) ≤
(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
J. Inequal. Pure and Appl. Math., 7(3) Art. 87, 2006
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A N I NEQUALITY FOR THE C LASS N UMBER
3
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
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,
and
1
γ = −2
2
(1.4)
Z
1
∞
ψ2 (t)
dt
t3
and
∞
Z
(1.5)
γ1 = −
1
2 log t − 3
ψ2 (t) dt.
t3
2. R ESULTS
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)!
J. Inequal. Pure and Appl. Math., 7(3) Art. 87, 2006
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4
O LIVIER B ORDELLÈS
3. T HE C ASE 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
2
+ (3γ − 1) log x + 3γ − 3γ − 3γ1 + 1 + R (x)
S3 (x) = x
2
where:
|R (x)| ≤ 2.36 x2/3 log x.
The proof of this lemma needs some technical results:
Lemma 3.3. Let x, y ≥ 1 be real numbers.
(i) If e3/2 ≤ y ≤ x, then we have:
X1
k≤y
k
log
x
k
(log y)2
= log x log y −
+ γ log x − γ1 + R1 (x, y)
2
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
with:
|R3 (y)| ≤
J. Inequal. Pure and Appl. Math., 7(3) Art. 87, 2006
1
y 1/2
1
+ .
y
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A N I NEQUALITY FOR THE C LASS N UMBER
5
Proof. (i) By the Euler-MacLaurin summation formula, we get:
x
X1
log
k
k
k≤y
Z y
x
log x
1
ψ (y)
x
=
+
log
dt −
log
2
t
y
y
1 t
Z y
x
ψ2 (y)
2 log (x/t) + 3
−
log
+
1
−
ψ2 (t) dt
y2
y
t3
1
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:
x
X1
(log y)2
log
= log x log y −
+ γ log x − γ1 + R1 (x, y)
k
k
2
k≤y
and since e3/2 ≤ y ≤ x, we have:
log (x/y) log (x/y) + 1 log x log y − 1
+
+
+
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
|R1 (x, y)| ≤
which concludes the proof of Lemma 3.3.
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
x 1 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
J. Inequal. Pure and Appl. Math., 7(3) Art. 87, 2006
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6
O LIVIER B ORDELLÈS
with
r
x 1
+
n 2
r
x
x
x
x
x 1
+
|R5 (x, T )| ≤ S2
≤ log
+ (2γ − 1) +
T
2
T
T
T
T
|R4 (x, n)| ≤
and hence:
(
S3 (x) = x log x log T −
(log T )2
+ γ log x
2
o
− γ1 + R6 (x, T ) + (2γ − 1) (log T + γ + R7 (T ))
)
(
x
X
(log (x/T ))2
2
+
R4 (x, n) + x
+ 2γ log
+ γ − 2γ1 + R8 (x, T )
2
T
n≤T
x
+ R5 (x, T ) − x log
− (2γ − 1) x − T R9 (x, T )
T
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
x
r
x 1
|R9 (x, T )| ≤
+
T
2
|R6 (x, T )| ≤
and thus:
(
S3 (x) = x
)
(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 )
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
)
(log x)2
+ (3γ − 1) log x + 3γ 2 − 3γ − 3γ1 + 1 + R11 (x, T )
2
J. Inequal. Pure and Appl. Math., 7(3) Art. 87, 2006
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A N I NEQUALITY FOR THE C LASS N UMBER
7
with:
|R11 (x, T )| ≤
√
√
x log (x/T ) x log x
+
+ 4 xT − x
2
2T
4T
r
x
x
1
2x
x
+
log
+ 2 (2γ − 1) +
+ 2T + .
T
T
T
T
2
We choose:
T = x1/3 ,
which gives:
(
S3 (x) = x
)
(log x)2
+ (3γ − 1) log x + 3γ 2 − 3γ − 3γ1 + 1 + R12 (x) ,
2
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.
4. P ROOF OF T HEOREM 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
un =
n3 × n!
.
4 (n + 1/2)n
We get:
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
and hence (un ) is decreasing, and thus:
un ≤ u3 =
which concludes the proof of Lemma 4.1.
J. Inequal. Pure and Appl. Math., 7(3) Art. 87, 2006
324
≤ 1,
343
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8
O LIVIER B ORDELLÈS
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
x
t Sn (t) dt + x
t−2 Sn (t) dt
≤ Sn (x) + x
1
13
(
)
n
Z x
n−1
(log x + n − 2)
1
1
1
(log t + n − 2)n−1
≤x
+
n+
+
dt
(n − 1)!
n!
2
(n − 1)! 13
t
(
n
)
(log x + n − 2)n (log x + n − 2)n−1
1
1
n
+
+
n+
− n + log 13e−2
=x
n!
(n − 1)!
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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edu.au/article.php?sid=190]
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244.
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at s = 1, and relative class number, J. Number Theory, 85 (2000), 263–282.
[4] D.S. MITRINOVIĆ, J. SÁNDOR
Verlag, 2nd printing, (2005).
AND
B. CRSTICI, Handbook of Number Theory I, Springer-
[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 Regulator-abschätzung, Fakultät für
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J. Inequal. Pure and Appl. Math., 7(3) Art. 87, 2006
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