Journal of Inequalities in Pure and
Applied Mathematics
AN EXPLICIT MERTENS’ TYPE INEQUALITY FOR ARITHMETIC
PROGRESSIONS
volume 6, issue 3, article 67,
2005.
OLIVIER BORDELLÈS
2 allée de la combe
La Boriette
43000 Aiguilhe, France.
Received 27 September, 2004;
accepted 04 May, 2005.
Communicated by: J. Sándor
EMail: borde43@wanadoo.fr
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
172-04
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Abstract
We give an explicit Mertens type formula for primes in arithmetic progressions
using mean values of Dirichlet L-functions at s = 1.
2000 Mathematics Subject Classification: 11N13, 11M20
Key words: Mertens’ formula, Arithmetic progressions, Mean values of Dirichlet
L−functions.
Contents
1
2
3
4
Introduction and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Sums with Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
The Polyá-Vinogradov Inequality and Character Sums with
Primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5
Mean Value Estimates of Dirichlet L−functions . . . . . . . . . . . 12
6
Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
References
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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1.
Introduction and Main Result
The very useful Mertens’ formula states that
Y
1
1
e−γ
=
1+O
1−
p
log x
log x
p≤x
for any real number x ≥ 2, where γ ≈ 0.577215664 . . . denotes the Euler
constant. Some explicit inequalities have been given in [4] where it is showed
for example that
−1
Y
1
(1.1)
< eγ δ (x) log x,
1−
p
p≤x
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
where
1
δ (x) := 1 +
.
(log x)2
(1.2)
Let 1 ≤ l ≤ k be positive integers satisfying (k, l) = 1. The aim of this
paper is to provide an explicit upper bound for the product
−1
Y 1
(1.3)
1−
.
p
p≤x
p≡l (mod k)
In [2, 5], the authors gave asymptotic formulas for (1.3) in the form
Y 1
1−
∼ c (k, l) (log x)−1/ϕ(k) ,
p
p≤x
p≡l (mod k)
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where ϕ is the Euler totient function and c (k, l) is a constant depending on l
and k. Nevertheless, because of the non-effectivity of the Siegel-Walfisz theorem, one cannot compute the implied constant in the error term. Moreover,
the constant c (k, l) is given only for some particular cases in [2], whereas K.S.
Williams established a quite complicated expression of c (k, l) involving a product of Dirichlet L−functions L (s; χ) and a function K (s; χ) at s = 1, where
K (s; χ) is the generating Dirichlet series of the completely multiplicative function kχ defined by
(
−χ(p) )
1
χ (p)
kχ (p) := p 1 − 1 −
1−
p
p
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
for any prime number p and any Dirichlet character χ modulo k. The author
then gave explicit expressions of c (k, l) in the case k = 24.
It could be useful to have an explicit upper bound for (1.3) valid for a large
range of k and x. Indeed, we shall see in a forthcoming paper that such a bound
could be used to estimate class numbers of certain cyclic number fields. We
prove the following result:
Theorem 1.1. Let 1 ≤ l ≤ k be positive integers satisfying (k, l) = 1 and
k ≥ 37, and x be a positive real number such that x > k. We have:
Y
p≤x
p≡l (mod k)
1−
1
p
−1
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v
1
γ
ϕ(k)
u
Y
e ϕ (k)
1
2(γ−B) u
tζ (2)
<e
1− 2 ·
log x
· Φ (x, k) ,
p
k
p|k
where
(
Φ (x, k) := exp
2
log x
!)
√
√
2 k log k X L 0
(1; χ) + 2 k log k + E − γ
,
ϕ (k) χ6=χ L
0
B ≈ 0.261497212847643 . . . and E ≈ 1.332582275733221 . . .
The restriction k ≥ 37 is given here just to use a simpler expression of the
Polyá-Vinogradov inequality, but one can prove a similar result with k ≥ 9 only,
the constants in Φ (x, k) being slightly larger.
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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2.
Notation
p denotes always a prime, 1 ≤ l ≤ k are positive integers satisfying (k, l) = 1
and k ≥ 37, x > k is a real number,
!
n
X
1
γ := lim
− log n ≈ 0.5772156649015328 . . .
n→∞
k
k=1
is the Euler constant and
γ1 := lim
n→∞
n
X
log k
k=1
k
(log n)2
−
2
!
≈ −0.07281584548367 . . .
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
is the first Stieltjes constant. Similarly,
E := lim
n→∞
log n −
X log p
p≤n
!
B := lim
n→∞
X1
p≤n
p
≈ 1.332582275733221 . . .
p
and
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!
− log log n
≈ 0.261497212847643 . . .
χ denotes a Dirichlet character modulo k and χ0 is the principal character modulo k. For anyP
Dirichlet character χ modulo k and any s ∈ C such that ReP
s > 1,
∞ χ(n)
L (s; χ) :=
n=1 ns is the Dirichlet L− function associated to χ.
χ6=χ0
means that the sum is taken over all non-principal characters modulo k. Λ is
the Von Mangoldt function and f ∗ g denotes the usual Dirichlet convolution
product.
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3.
Sums with Primes
From [4] we get the following estimates:
Lemma 3.1.
X
p
∞
X
1
log p
=E−γ
pα
α=2
and
∞
XX
1
= γ − B.
αpα
p α=2
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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4.
The Polyá-Vinogradov Inequality and
Character Sums with Primes
Lemma 4.1. Let χ be any non-principal Dirichlet character modulo k ≥ 37.
(i) For any real number x ≥ 1,
X
9√
k log k.
χ (n) <
10
n≤x
(ii) Let F ∈ C 1 ([1; +∞[ , [0; +∞[) such that F (t) & 0. For any real numt→∞
ber x ≥ 1,
X
9
√
χ (n) F (n) ≤ F (x) k log k.
n>x
5
(iii) For any real number x > k,
0
X χ (p) √
L
2
2 k log k (1; χ) + 1 + E − γ .
<
log x
L
p
p>x
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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Proof.
(i) The result follows from Qiu’s improvement of the Polyá-Vinogradov inequality (see [3, p. 392]).
(ii) Abel summation and (i).
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(iii) Let χ 6= χ0 be a Dirichlet character modulo k ≥ 37 and x > k be any real
number.
(
1, if n = 1
(a) Since χ (µ ∗ 1) = ε where ε (n) =
and 1 (n) = 1,
0, otherwise
we get:
X µ (d) χ (d) X χ (m)
=1
d
m
d≤x
m≤x/d
and hence, since χ 6= χ0 ,

X µ (d) χ (d)
d≤x
d

X
X
µ (d) χ (d)
χ (m)
1

=
+ 1
L (1; χ) d≤x
d
m
Olivier Bordellès
m>x/d
and thus, using (ii),
√
√
X µ (d) χ (d) 9
k
log
k
+
1
2
k log k
(4.1)
<
.
≤ 5
d
|L (1; χ)|
|L (1; χ)|
d≤x
(b) Since log = Λ ∗ 1, we get:
X χ (n) Λ (n)
n≤x
An Explicit Mertens’ Type
Inequality for Arithmetic
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n
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X µ (d) χ (d) X χ (m) log m
d
m
d≤x
m≤x/d


X
X
X χ (m) log m

=
+
m
=
d≤x/e
x/e<d≤x
m≤x/d
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=
X µ (d) χ (d) X χ (m) log m χ (2) log 2
+
d
m
2
d≤x/e
= −L 0 (1; χ)
m≤x/d
x/e<d≤x
µ (d) χ (d)
d
X µ (d) χ (d)
X µ (d) χ (d) X χ (m) log m
−
d
d
m
d≤x/e
+
X
χ (2) log 2
2
d≤x/e
X
x/e<d≤x
m>x/d
µ (d) χ (d)
d
and, by using (ii), (4.1) and the trivial bound for the third sum, we
get:
X χ (n) Λ (n) (4.2) n
n≤x
0
√
L
< k log k 2 (1; χ)
L

X
9
x
log 2
e 
+
log
+
1+
5x
d
2
x 
d≤x/e
0
√
18 log 2 L
e
+
1+
≤ k log k 2 (1; χ) +
L
5e
2
37
0
√
L
< 2 k log k (1; χ) + 1
L
since x > q ≥ 37.
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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(c) By Abel summation, we get:
X χ (p) log p X χ (p) 2
max ≤
.
log x t≥x p
p
p>x
p≤t
Moreover,
X χ (p) log p
p≤t
p
=
X χ (n) Λ (n)
n≤t
n
−
∞
XX
χ (pα ) log p
p α=2
pα ≤t
pα
and then:
∞
X χ (p) log p X χ (n) Λ (n) X
X
1
≤
+
log
p
p
n
pα
p
α=2
p≤t
n≤t
X χ (n) Λ (n) =
+E−γ
n
n≤t
0
√
L
< 2 k log k (1; χ) + 1 + E − γ
L
by (4.2) . This concludes the proof of Lemma 4.1.
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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Mean Value Estimates of Dirichlet L−functions
5.
Lemma 5.1.
(i) For any positive integers j, k,


jk

X
X
log p  c0 (j, k) 2ω(k)
1
ϕ (k)
=
log (jk) + γ +
+
n
k 
p − 1
jk
n=1
p|k
(n,k)=1
where ω (k) :=
P
p|k
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
1 and |c0 (j, k)| ≤ 1.
(ii) For any positive integer k ≥ 9,
Olivier Bordellès
k
ϕ (k)
2
ζ (2)
Y
p|k
1−
2

X log p
1
π2 
 ≤ 0.
+2γ
+γ+
−
log
k
+
1
p2
3
p−1
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p|k
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(iii) For any positive integer k ≥ 9,
Y
|L (1; χ)|1/ϕ(k)
χ6=χ0
1
Y
p
1 2
≤ ζ (2)
1− 2
.
p
p|k
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jk
X
1 X µ (d) X 1 X µ (d)
jk
ε (d) d
=
=
log
+γ+
n
d
n
d
d
jk
n=1
(n,k)=1
d|k
n≤jk/d
d|k
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where |ε (d)| ≤ 1 and hence:
jk
X
X µ (d) X µ (d) log d 1 X
1
= {log (jk) + γ}
−
+
ε (d) µ (d)
n
d
d
jk
n=1
d|k
(n,k)=1
d|k
d|k
and we conclude by noting that
X µ (d)
d
d|k
X µ (d) log d
d
d|k
=
=−
ϕ (k)
,
k
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
ϕ (k) X log p
k
p−1
Olivier Bordellès
p|k
Title Page
and
X
X
ε (d) µ (d) ≤
µ2 (d) = 2ω(k) .
d|k
d|k
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(ii) Define
A (k) :=
k
ϕ (k)
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2
ζ (2)
Y
p|k
1
1− 2
p
Close
2

+ 2γ1 + γ +
X log p
π2 
 .
− log k +
3
p−1
p|k
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Using [1] we check the inequality for 9 ≤ k ≤ 513 and then suppose
k ≥ 514. Since
Y p
Y 1
k
p p−1
=
≤
ϕ (k)
p−1
p|k
p|k
we have taking logarithms
X log p
k
k
≥ log
≥ log
p−1
ϕ (k)
k−1
p|k
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
and from the inequality ([4])
2.50637
k
< eγ log log k +
ϕ (k)
log log k
Olivier Bordellès
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valid for any integer k ≥ 3, we obtain
Contents
2
2.50637
log log k
2 2
π2
k
+ 2γ1 + γ +
− log
<0
3
k−1
A (k) ≤ ζ (2) eγ log log k +
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if k ≥ 514.
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(iii) First,
X
χ6=χ0
|L (1; χ)|2 = lim S (N )
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where
S (N ) :=

Nk
X
χ (n) χ (m) 
−
nm
m,n=1
Nk
X
n=1
(n,k)=1
2
1
 .
n
Following a standard argument, we have using (i):
S (N )
= ϕ (k)

Nk
X
m6=n=1
m≡n (mod k)
(n,k)=(m,k)=1
= ϕ (k)
1

−
mn
Nk
X
1
+ ϕ (k)
n2
n=1
(n,k)=1
≤ ϕ (k) ζ (2)
Y
p|k
+ 2ϕ (k)
1
1− 2
p
−j)k
N (N
X
X
j=1
n=1
(n,k)=1
Nk
X
n=1
(n,k)=1
2
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
1

n
Nk
X
m6=n=1
m≡n (mod k)
(n,k)=(m,k)=1
Olivier Bordellès
2

Nk
1
 X 1
−

mn
n
n=1
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(n,k)=1
JJ
J

1

−
n (n + jk)
Nk
X
n=1
(n,k)=1
2
1

n
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1
= ϕ (k) ζ (2)
1− 2
p
p|k

 
2
(N −j)k
N
k
N
N
k
X 1  X 1
2ϕ (k) X 1  X 1
+
−

−

k j=1 j
n n=1+jk n
n
n=1
n=1
Y
(n,k)=1
≤ ϕ (k) ζ (2)
Y
1−
p|k
= ϕ (k) ζ (2)
Y
1−
p|k
1
p2
1
p2
+
(n,k)=1
(n,k)=1
2ϕ (k)
k
N
X
j=1
jk

1 X 1 
−
j n=1 n
(n,k)=1
Nk
X
n=1
(n,k)=1
2
1

n
Olivier Bordellès





ω(k)
X
log p
c0 (j, k) 2
 ϕ (k) log (jk) + γ +

+
+
k j=1 j
k 
p − 1
jk
p|k



2


ω(k)
X
ϕ (k)
log p
c0 (N, k) 2
 .
−
log (N k) + γ +
+
k 
p − 1
Nk
N
2ϕ (k) X 1
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions

p|k
We now neglect the dependance of c0 in k. Since
M
X
1
c1 (M )
= log M + γ +
m
M
m=1
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and
M
X
(log M )2
c2 (M ) log M
log m
=
+ γ1 +
,
m
2
M
m=1
where 0 < c1 (M ) ≤
1
2
and |c2 (M )| ≤ 1, we get:
Y
1
S (N ) ≤ ϕ (k) ζ (2)
1− 2
p
p|k
2 ϕ (k)
2c2 (N ) log N
+
(log N )2 + 2γ1 +
k
N


X log p
c
(N
)
1
 log N + γ +
+ 2 log k + γ +
p−1
N
p|k

2 

X log p


− log (N k) + γ +
p−1 
p|k
( N
2ω(k)+1 ϕ (k) X c0 (j)
+
k2
j2
j=1


X
c0 (N ) 
log p  22ω(k) c20 (N )
log (N k) + γ +
−
−
N
p−1 
N 2k2
An Explicit Mertens’ Type
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Olivier Bordellès
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p|k
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1
= ϕ (k) ζ (2)
1− 2
p
p|k


2
2 
X
ϕ (k)
log p 
+
2γ1 + γ − log k +

k
p−1
p|k



X
2c1 (N ) 
log p  2c2 (N ) log N 
+
log k + γ +
+

N
p−1
N
p|k
( N
2ω(k)+1 ϕ (k) X c0 (j)
+
k2
j2
j=1


X
log p  22ω(k) c20 (N )
c0 (N ) 
log (N k) + γ +
−
−
N
p−1 
N 2k2
Y
p|k
lim S (N ) ≤ ϕ (k) ζ (2)
N →∞
p|k
+
1
1− 2
p

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Y
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
2 
2 
X
ϕ (k)
log p  
2γ1 + γ − log k +

k
p−1 
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p|k
ω(k)
+
2
ϕ (k) π
3k 2
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2
J. Ineq. Pure and Appl. Math. 6(3) Art. 67, 2005
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and the inequality 2ω(k) ≤ ϕ (k) (valid for any integer k ≥ 3 and 6= 6)
implies
Y
1
lim S (N ) ≤ ϕ (k) ζ (2)
1− 2
N →∞
p
p|k


2 
2 
2
X
π
log p  
ϕ (k)
+
2γ1 + γ +
− log k +

k
3
p−1 
p|k
Y
1
= (ϕ (k) − 1) ζ (2)
1− 2
p
p|k

2 2
Y
ϕ (k) 
k
1
+
ζ (2)
1− 2
 ϕ (k)
k
p
p|k

2 
2
X
π
log p  

+2γ1 + γ +
− log k +
3
p−1 
p|k
Y
1
≤ (ϕ (k) − 1) ζ (2)
1− 2
p
p|k
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if k ≥ 9 by (ii). Hence
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X
Y
1
1
2
|L (1; χ)| ≤ ζ (2)
1− 2 .
ϕ (k) − 1 χ6=χ
p
0
p|k
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Now the IAG inequality implies:
(
)
Y
X
1
1
|L (1; χ)| ϕ(k) = exp
log |L (1; χ)|2
2ϕ
(k)
χ6=χ0
χ6=χ0
(
!)
X
ϕ (k) − 1
1
≤ exp
log
|L (1; χ)|2
2ϕ (k)
ϕ (k) − 1 χ6=χ
0

≤ ζ (2)
Y
p|k

≤ ζ (2)
Y
p|k
 ϕ(k)−1
2ϕ(k)
1 
1− 2
p
1−
 12
1 
.
p2
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6.
Proof of the Theorem
Lemma 6.1. If χ0 is the principal character modulo k and if x > k, then:
Y
p≤x
1
1−
p
−χ0 (p)
<
eγ ϕ (k) δ (x)
· log x,
k
where δ is the function defined in (1.2).
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Proof. Since x > k,
Y
p≤x
p|k
1
1−
p
=
Y
p|k
1
1−
p
=
ϕ (k)
k
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and then
Y
p≤x
1−
1
p
−χ0 (p)
=
Y
p≤x
p-k
=
Y
p≤x
=
and we use (1.1) .
1−
1
p
1
1−
p
Contents
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J
−1 Y Y
p≤x
p|k
ϕ (k)
1
1−
k p≤x
p
1
1−
p
−1
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Proof of the theorem. Let 1 ≤ l ≤ k be positive integers satisfying (k, l) = 1
and k ≥ 37, and x be a positive real number such that x > k. We have:
−ϕ(k) Y −χ0 (p) Y Y −χ(p) !χ(l)
Y 1
1
1
1−
=
1−
·
1−
p
p
p
p≤x
p≤x
p≤x
χ6=χ
0
p≡l (mod k)
:= Π1 × Π2
eγ ϕ (k) δ (x)
with Π1 <
· log x by Lemma 6.1. Moreover,
k
(
!)
X
X
1
Π2 = exp
χ (l) −
χ (p) log 1 −
p
p≤x
χ6=χ0
!
∞
X
XX
χ (p)
= exp
χ (l)
αpα
p≤x α=1
χ6=χ0
!)
(
∞
X
X χ (p) X X
χ (p)
+
= exp
χ (l)
p
αpα
p≤x
p≤x α=2
χ6=χ
An Explicit Mertens’ Type
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and if χ 6= χ0 , we have
L (1; χ) =
Y
p
= exp
II
I
χ (p)
1−
p
X χ (p)
p≤x
p
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−1
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+
X χ (p)
p>x
p
+
∞
XX
χ (pα )
p
α=2
αpα
!
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J. Ineq. Pure and Appl. Math. 6(3) Art. 67, 2005
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and thus
Π2 =
Y
L (1; χ)χ(l)
χ6=χ0
(
× exp
X
χ (l) −
X χ (p)
p>x
χ6=χ0
p
+
∞
XX
χ (p)
p≤x α=2
αpα
−
∞
XX
χ (pα )
p
α=2
!)
αpα
and hence
)
∞
XX
Y
X X χ (p) 1
|Π2 | ≤
|L (1; χ)| · exp
+ 2 (ϕ (k) − 1)
p αpα
p α=2
χ6=χ0
χ6=χ0 p>x
!)
(
Y
X X χ (p) = e2(ϕ(k)−1)(γ−B)
|L (1; χ)| · exp
p
p>x
(
χ6=χ0
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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χ6=χ0
Contents
and we use Lemma 4.1 (iii) and Lemma 5.1 (iii). We conclude the proof by
δ(x)
< 1.
noting that, if x > 37, e2(γ−B)
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References
[1] PARI/GP, Available by anonymous ftp from the URL: ftp://megrez.
math.u-bordeaux.fr/pub/pari.
[2] E. GROSSWALD, Some number theoretical products, Rev. Colomb. Mat.,
21 (1987), 231–242.
[3] D.S. MITRINOVIĆ AND J. SÁNDOR (in cooperation with B. CRSTICI),
Handbook of Number Theory, Kluwer Acad. Publishers, ISBN: 0-79233823-5.
[4] J.B. ROSSER AND L. SCHŒNFELD, Approximate formulas for some
functions of prime numbers, Illinois J. Math., 6 (1962), 64–94.
[5] K.S. WILLIAMS, Mertens’ theorem for arithmetic progressions, J. Number
Theory, 6 (1974), 353–359.
An Explicit Mertens’ Type
Inequality for Arithmetic
Progressions
Olivier Bordellès
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