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volume 6, issue 3, article 67, 2005.

Received 27 September, 2004;

accepted 04 May, 2005.

Communicated by:J. Sándor

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Journal of Inequalities in Pure and Applied Mathematics

AN EXPLICIT MERTENS’ TYPE INEQUALITY FOR ARITHMETIC PROGRESSIONS

OLIVIER BORDELLÈS

2 allée de la combe La Boriette

43000 Aiguilhe, France.

EMail:borde43@wanadoo.fr

c

2000Victoria University ISSN (electronic): 1443-5756 172-04

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An Explicit Mertens’ Type Inequality for Arithmetic

Progressions Olivier Bordellès

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J. Ineq. Pure and Appl. Math. 6(3) Art. 67, 2005

http://jipam.vu.edu.au

Abstract

We give an explicit Mertens type formula for primes in arithmetic progressions using mean values of Dirichlet L-functions ats= 1.

2000 Mathematics Subject Classification:11N13, 11M20

Key words: Mertens’ formula, Arithmetic progressions, Mean values of Dirichlet L−functions.

1. Introduction and Main Result

The very useful Mertens’ formula states that Y

p≤x

1− 1

p

= e^−γ logx

1 +O

1 logx

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

(1.1) Y

p≤x

1−1

p −1

< e^γδ(x) logx, where

(1.2) δ(x) := 1 + 1

(logx)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

(1.3) Y

p≤x p≡l(modk)

1−1

p ⁻¹

.

In [2,5], the authors gave asymptotic formulas for(1.3)in the form Y

p≤x p≡l(modk)

1− 1

p

∼c(k, l) (logx)^−1/ϕ(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 constantc(k, l)is given only for some particular cases in [2], whereas K.S.

Williams established a quite complicated expression ofc(k, l)involving a product of Dirichlet L−functionsL(s;χ)and a function K(s;χ)at s = 1, where K(s;χ)is the generating Dirichlet series of the completely multiplicative func- tionkχdefined by

k_χ(p) :=p (

1−

1−χ(p)

p 1− 1

p

−χ(p))

for any prime number p and any Dirichlet character χ modulo k. The author then gave explicit expressions ofc(k, l)in the casek = 24.

It could be useful to have an explicit upper bound for(1.3)valid for a large range ofkandx.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,andxbe a positive real number such thatx > k.We have:

Y

p≤x p≡l(modk)

1− 1

p −1

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< e^2(γ−B) v u u

tζ(2)Y

p|k

1− 1

p²

·

e^γϕ(k) k logx

_ϕ(k)¹

·Φ (x, k),

where

Φ (x, k) := exp ( 2

logx 2√

klogk ϕ(k)

X

χ6=χ₀

L⁰ L (1;χ)

+ 2√

klogk+E−γ

!) ,

B ≈0.261497212847643. . .andE ≈1.332582275733221. . .

The restrictionk ≥ 37is given here just to use a simpler expression of the Polyá-Vinogradov inequality, but one can prove a similar result withk ≥9only, the constants inΦ (x, k)being slightly larger.

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2. Notation

pdenotes always a prime,1 ≤ l ≤ k are positive integers satisfying(k, l) = 1 andk≥37, x > kis a real number,

γ := lim

n→∞

n

X

k=1

1

k −logn

!

≈0.5772156649015328. . .

is the Euler constant and γ₁ := lim

n→∞

n

X

k=1

logk

k − (logn)² 2

!

≈ −0.07281584548367. . . is the first Stieltjes constant. Similarly,

E := lim

n→∞ logn−X

p≤n

logp p

!

≈1.332582275733221. . . and

B := lim

n→∞

X

p≤n

1

p −log logn

!

≈0.261497212847643. . .

χdenotes a Dirichlet character modulokandχ₀ is the principal character modulok.For any Dirichlet characterχmodulokand anys ∈Csuch thatRes >1, L(s;χ) := P∞

n=1 χ(n)

n^s is the Dirichlet L− function associated to χ. P

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

logp

∞

X

α=2

1

p^α =E−γ and X

p

∞

X

α=2

1

αp^α =γ−B.

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4. The Polyá-Vinogradov Inequality and Character Sums with Primes

Lemma 4.1. Letχbe any non-principal Dirichlet character modulok≥37.

(i) For any real numberx≥1,

X

n≤x

χ(n)

< 9 10

√

klogk.

(ii) LetF ∈C¹([1; +∞[, [0; +∞[)such thatF(t) &

t→∞

0.For any real num- berx≥1,

X

n>x

χ(n)F (n)

≤ 9

5F (x)√

klogk.

(iii) For any real numberx > k,

X

p>x

χ(p) p

< 2 logx

2√

klogk

L⁰ L (1;χ)

+ 1

+E−γ

.

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) Letχ6=χ₀be a Dirichlet character modulok ≥37andx > kbe any real number.

(a) Sinceχ(µ∗1) =εwhereε(n) =

( 1, ifn= 1

0, otherwise and1(n) = 1, we get:

X

d≤x

µ(d)χ(d) d

X

m≤x/d

χ(m) m = 1 and hence, sinceχ6=χ₀,

X

d≤x

µ(d)χ(d)

d = 1

L(1;χ)



 X

d≤x

µ(d)χ(d) d

X

m>x/d

χ(m) m + 1





and thus, using (ii), (4.1)

X

d≤x

µ(d)χ(d) d

≤

9 5

√klogk+ 1

|L(1;χ)| < 2√ klogk

|L(1;χ)|. (b) Sincelog = Λ∗1, we get:

X

n≤x

χ(n) Λ (n)

n =X

d≤x

µ(d)χ(d) d

X

m≤x/d

χ(m) logm m

=



 X

d≤x/e

+ X

x/e<d≤x



 X

m≤x/d

χ(m) logm m

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= X

d≤x/e

µ(d)χ(d) d

X

m≤x/d

χ(m) logm

m + χ(2) log 2 2

X

x/e<d≤x

µ(d)χ(d) d

=−L⁰(1;χ) X

d≤x/e

µ(d)χ(d)

d − X

d≤x/e

µ(d)χ(d) d

X

m>x/d

χ(m) logm m + χ(2) log 2

2

X

x/e<d≤x

µ(d)χ(d) d

and, by using (ii), (4.1)and the trivial bound for the third sum, we get:

X

n≤x

χ(n) Λ (n) n

(4.2)

<√ klogk

2

L⁰ L (1;χ)

+ 9 5x

X

d≤x/e

logx d

+ log 2 2

1 + e

x







≤√ klogk

2

L⁰ L (1;χ)

+18

5e + log 2 2

1 + e

37

<2√ klogk

L⁰ L (1;χ)

+ 1

sincex > q ≥37.

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(c) By Abel summation, we get:

X

p>x

χ(p) p

≤ 2

logxmax

t≥x

X

p≤t

χ(p) logp p

.

Moreover, X

p≤t

χ(p) logp

p =X

n≤t

χ(n) Λ (n)

n −X

p

∞

X

α=2 p^α≤t

χ(p^α) logp p^α

and then:

X

p≤t

χ(p) logp p

≤

X

n≤t

χ(n) Λ (n) n

+X

p

logp

∞

X

α=2

1 p^α

=

X

n≤t

χ(n) Λ (n) n

+E−γ

<2√ klogk

L⁰ L (1;χ)

+ 1

+E−γ by(4.2).This concludes the proof of Lemma4.1.

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5. Mean Value Estimates of Dirichlet L−functions

Lemma 5.1.

(i) For any positive integersj, k,

jk

X

n=1 (n,k)=1

1

n = ϕ(k) k







log (jk) +γ +X

p|k

logp p−1







+ c₀(j, k) 2^ω(k) jk whereω(k) :=P

p|k1and|c₀(j, k)| ≤1.

(ii) For any positive integerk ≥9, k

ϕ(k) 2

ζ(2)Y

p|k

1− 1

p²

+2γ₁+γ+π² 3 −



logk+X

p|k

logp p−1





2

≤0.

(iii) For any positive integerk ≥9, Y

χ6=χ₀

|L(1;χ)|^1/ϕ(k) ≤p

ζ(2)Y

p|k

1− 1

p² ¹₂

.

Proof.

(i)

jk

X

n=1 (n,k)=1

1

n =X

d|k

µ(d) d

X

n≤jk/d

1

n =X

d|k

µ(d) d

log

jk d

+γ+ε(d)d jk

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where|ε(d)| ≤1and hence:

jk

X

n=1 (n,k)=1

1

n ={log (jk) +γ}X

d|k

µ(d) d −X

d|k

µ(d) logd

d + 1

jk X

d|k

ε(d)µ(d)

and we conclude by noting that X

d|k

µ(d)

d = ϕ(k) k , X

d|k

µ(d) logd

d =−ϕ(k) k

X

p|k

logp p−1

and

X

d|k

ε(d)µ(d)

≤X

d|k

µ²(d) = 2^ω(k).

(ii) Define

A(k) :=

k ϕ(k)

2

ζ(2)Y

p|k

1− 1

p²

+ 2γ₁ +γ+ π² 3 −



logk+X

p|k

logp p−1





2

.

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Using [1] we check the inequality for 9 ≤ k ≤ 513 and then suppose k≥514.Since

k

ϕ(k) =Y

p|k

p

p−1 ≤Y

p|k

p^p−1¹ we have taking logarithms

X

p|k

logp p−1 ≥log

k ϕ(k)

≥log k

k−1

and from the inequality ([4]) k

ϕ(k) < e^γlog logk+ 2.50637 log logk valid for any integerk ≥3,we obtain

A(k)≤ζ(2)

e^γlog logk+ 2.50637 log logk

2

+ 2γ₁+γ+π² 3 −

log

k² k−1

2

<0 ifk ≥514.

(iii) First,

X

χ6=χ₀

|L(1;χ)|² = lim

N→∞S(N)

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where

S(N) :=

N k

X

m,n=1

χ(n)χ(m)

nm −







N k

X

n=1 (n,k)=1

1 n







2

.

Following a standard argument, we have using (i):

S(N)

=ϕ(k)

N k

X

m6=n=1 m≡n(modk) (n,k)=(m,k)=1

1 mn−







N k

X

n=1 (n,k)=1

1 n







2

=ϕ(k)

N k

X

n=1 (n,k)=1

1

n² +ϕ(k)

N k

X

m6=n=1 m≡n(modk) (n,k)=(m,k)=1

1 mn −







N k

X

n=1 (n,k)=1

1 n







2

≤ϕ(k)ζ(2)Y

p|k

1− 1

p²

+ 2ϕ(k)

N

X

j=1 (N−j)k

X

n=1 (n,k)=1

1

n(n+jk)−







N k

X

n=1 (n,k)=1

1 n







2

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=ϕ(k)ζ(2)Y

p|k

1− 1

p²

+2ϕ(k) k

N

X

j=1

1 j







(N−j)k

X

n=1 (n,k)=1

1 n −

N k

X

n=1+jk (n,k)=1

1 n





−







N k

X

n=1 (n,k)=1

1 n







2

≤ϕ(k)ζ(2)Y

p|k

1− 1

p²

+ 2ϕ(k) k

N

X

j=1

1 j

jk

X

n=1 (n,k)=1

1 n −







N k

X

n=1 (n,k)=1

1 n







2

=ϕ(k)ζ(2)Y

p|k

1− 1

p²

+2ϕ(k) k

N

X

j=1

1 j



 ϕ(k)

k







log (jk) +γ+X

p|k

logp p−1







+ c₀(j, k) 2^ω(k) jk





−



 ϕ(k)

k







log (N k) +γ+X

p|k

logp p−1







+c₀(N, k) 2^ω(k) N k





2

.

We now neglect the dependance ofc₀ ink.Since

M

X

m=1

1

m = logM +γ +c₁(M) M

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and

M

X

m=1

logm

m = (logM)²

2 +γ₁+ c₂(M) logM

M ,

where0< c₁(M)≤ ¹₂ and|c₂(M)| ≤1, we get:

S(N)≤ϕ(k)ζ(2)Y

p|k

1− 1

p²

+

ϕ(k) k

2

(logN)²+ 2γ₁+2c₂(N) logN N + 2



logk+γ+X

p|k

logp p−1





logN +γ+c₁(N) N

−



log (N k) +γ+X

p|k

logp p−1





2





+2^ω(k)+1ϕ(k) k²

( _N X

j=1

c₀(j) j²

−c₀(N) N



log (N k) +γ+X

p|k

logp p−1











− 2^2ω(k)c²₀(N) N²k²

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=ϕ(k)ζ(2)Y

p|k

1− 1

p²

+

ϕ(k) k

2





2γ₁+γ−



logk+X

p|k

logp p−1





2

+2c₁(N) N



logk+γ+X

p|k

logp p−1



+2c₂(N) logN N







+2^ω(k)+1ϕ(k) k²

( _N X

j=1

c₀(j) j²

−c₀(N) N



log (N k) +γ+X

p|k

logp p−1











− 2^2ω(k)c²₀(N) N²k² and then

N→∞lim S(N)≤ϕ(k)ζ(2)Y

p|k

1− 1

p²

+

ϕ(k) k

2





2γ₁+γ−



logk+X

p|k

logp p−1





2





+2^ω(k)ϕ(k)π² 3k²

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and the inequality2^ω(k) ≤ ϕ(k) (valid for any integerk ≥ 3 and 6= 6) implies

N→∞lim S(N)≤ϕ(k)ζ(2)Y

p|k

1− 1

p²

+

ϕ(k) k

2





2γ₁+γ+π² 3 −



logk+X

p|k

logp p−1





2





= (ϕ(k)−1)ζ(2)Y

p|k

1− 1

p²

+

ϕ(k) k

2



 k

ϕ(k) 2

ζ(2)Y

p|k

1− 1

p²

+2γ₁+γ+π² 3 −



logk+X

p|k

logp p−1





2





≤(ϕ(k)−1)ζ(2)Y

p|k

1− 1

p²

ifk ≥9by (ii). Hence 1 ϕ(k)−1

X

χ6=χ0

|L(1;χ)|² ≤ζ(2)Y

p|k

1− 1

p²

.

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Now the IAG inequality implies:

Y

χ6=χ₀

|L(1;χ)|^ϕ(k)¹ = exp ( 1

2ϕ(k) X

χ6=χ₀

log|L(1;χ)|² )

≤exp

(ϕ(k)−1

2ϕ(k) log 1 ϕ(k)−1

X

χ6=χ0

|L(1;χ)|²

!)

≤



ζ(2)Y

p|k

1− 1

p²





ϕ(k)−1 2ϕ(k)

≤



ζ(2)Y

p|k

1− 1

p²





1 2

.

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6. Proof of the Theorem

Lemma 6.1. Ifχ₀ is the principal character modulokand ifx > k,then:

Y

p≤x

1− 1

p

−χ0(p)

< e^γϕ(k)δ(x)

k ·logx, whereδis the function defined in(1.2).

Proof. Sincex > k, Y

p≤x p|k

1−1

p

=Y

p|k

1−1

p

= ϕ(k) k

and then

Y

p≤x

1−1

p

^−χ0(p)

= Y

p≤x p-k

1− 1

p ⁻¹

=Y

p≤x

1− 1

p −1

Y

p≤x p|k

1− 1

p

= ϕ(k) k

Y

p≤x

1−1

p −1

and we use(1.1).

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Proof of the theorem. Let 1 ≤ l ≤ k be positive integers satisfying(k, l) = 1 andk≥37,andxbe a positive real number such thatx > k.We have:

Y

p≤x p≡l(modk)

1− 1

p −ϕ(k)

=Y

p≤x

1−1

p

−χ0(p)

· Y

χ6=χ₀

Y

p≤x

1− 1

p

−χ(p)!χ(l)

:= Π₁×Π₂

withΠ₁ < e^γϕ(k)δ(x)

k ·logxby Lemma6.1. Moreover, Π₂ = exp

( X

χ6=χ0

χ(l) −X

p≤x

χ(p) log

1− 1 p

!)

= exp X

χ6=χ₀

χ(l)X

p≤x

∞

X

α=1

χ(p) αp^α

!

= exp (

X

χ6=χ0

χ(l) X

p≤x

χ(p)

p +X

p≤x

∞

X

α=2

χ(p) αp^α

!)

and ifχ6=χ₀,we have L(1;χ) = Y

p

1−χ(p) p

−1

= exp X

p≤x

χ(p)

p +X

p>x

χ(p)

p +X

p

∞

X

α=2

χ(p^α) αp^α

!

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and thus Π₂ = Y

χ6=χ0

L(1;χ)^χ(l)

×exp (

X

χ6=χ₀

χ(l) −X

p>x

χ(p)

p +X

p≤x

∞

X

α=2

χ(p) αp^α −X

p

∞

X

α=2

χ(p^α) αp^α

!)

and hence

|Π₂| ≤ Y

χ6=χ0

|L(1;χ)| ·exp (

X

χ6=χ0

X

p>x

χ(p) p

+ 2 (ϕ(k)−1)X

p

∞

X

α=2

1 αp^α

)

=e2(ϕ(k)−1)(γ−B) Y

χ6=χ0

|L(1;χ)| ·exp (

X

χ6=χ0

X

p>x

χ(p) p

!)

and we use Lemma 4.1 (iii) and Lemma 5.1 (iii). We conclude the proof by noting that, ifx >37, _e2(γ−B)^δ(x) <1.

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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. MITRINOVI ´C ANDJ. SÁNDOR (in cooperation with B. CRSTICI), Handbook of Number Theory, Kluwer Acad. Publishers, ISBN: 0-7923- 3823-5.

[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.