An Explicit Mertens' Type Inequality for Arithmetic Progressions - EMIS

Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 6, Issue 3, Article 67, 2005

AN EXPLICIT MERTENS’ TYPE INEQUALITY FOR ARITHMETIC PROGRESSIONS OLIVIER BORDELLÈS 2 ALLÉE DE LA COMBE L A B ORIETTE 43000 AIGUILHE [email protected]

Received 27 September, 2004; accepted 04 May, 2005 Communicated by J. Sándor

A BSTRACT. We give an explicit Mertens type formula for primes in arithmetic progressions using mean values of Dirichlet L-functions at s = 1. Key words and phrases: Mertens’ formula, Arithmetic progressions, Mean values of Dirichlet L−functions. 2000 Mathematics Subject Classification. 11N13, 11M20.

1. I NTRODUCTION AND M AIN R ESULT The very useful Mertens’ formula states that Y 1 e−γ 1 1− = 1+O 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) 1− < eγ δ (x) log x, p p≤x where (1.2)

δ (x) := 1 +

1 . (log x)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)

ISSN (electronic): 1443-5756 c 2005 Victoria University. All rights reserved.

172-04

2

O LIVIER B ORDELLÈS

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)

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) ) χ (p) 1 kχ (p) := p 1 − 1 − 1− p p 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: v 1 −1 γ ϕ(k) u Y Y 1 e ϕ (k) 1 2(γ−B) u tζ (2) 1− 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 . . .

is the first Stieltjes constant. Similarly, E := lim

n→∞

log n −

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

X log p p≤n

p

! ≈ 1.332582275733221 . . .

http://jipam.vu.edu.au/

A M ERTENS ’ INEQUALITY FOR A RITHMETIC

and B := lim

n→∞

X1 p≤n

p

3

PROGRESSIONS

! − log log n

≈ 0.261497212847643 . . .

χ denotes a Dirichlet character modulo k and χ0 is the principal character modulo k. For any P χ(n) Dirichlet character χ modulo k and any s ∈PC such that Re s > 1, L (s; χ) := ∞ n=1 ns is the Dirichlet L− function associated to χ. χ6=χ0 means that the sum is taken over all nonprincipal characters modulo k. Λ is the Von Mangoldt function and f ∗ g denotes the usual Dirichlet convolution product. 3. S UMS WITH P RIMES From [4] we get the following estimates: Lemma 3.1. X p

∞ X 1 =E−γ log p pα α=2

and

∞ XX 1 = γ − B. αpα p α=2

4. T HE P OLYÁ -V INOGRADOV I NEQUALITY AND C HARACTER S UMS WITH P RIMES Lemma 4.1. Let χ be any non-principal Dirichlet character modulo k ≥ 37. (i) For any real number x ≥ 1, X 9√ χ (n) < k log k. 10 n≤x (ii) Let F ∈ C 1 ([1; +∞[ , [0; +∞[) such that F (t) & 0. For any real number x ≥ 1, t→∞ X 9 √ χ (n) F (n) ≤ F (x) k log k. 5 n>x (iii) For any real number x > k, 0 X χ (p) √ L 2 2 k log k (1; χ) + 1 + E − γ . < log x p L p>x Proof. (i) The result follows from Qiu’s improvement of the Polyá-Vinogradov inequality (see [3, p. 392]). (ii) Abel summation and (i). (iii) Let χ 6= χ0 be a Dirichlet character modulo k ≥ 37 and x > k be any real number. 1, if n = 1 and 1 (n) = 1, we get: (a) Since χ (µ ∗ 1) = ε where ε (n) = 0, otherwise 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 1 µ (d) χ (d) χ (m)  = + 1 L (1; χ) d≤x d m


m>x/d


4


and thus, using (ii), √ √ X µ (d) χ (d) 9 k log k + 1 2 k log k 5 < . ≤ |L (1; χ)| |L (1; χ)| d d≤x

(4.1)

(b) Since log = Λ ∗ 1, we get: X χ (n) Λ (n) n≤x

n

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/ex p log x t≥x p 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


A M ERTENS ’ INEQUALITY FOR A RITHMETIC

5

PROGRESSIONS

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. 5. M EAN VALUE E STIMATES OF D IRICHLET L−FUNCTIONS Lemma 5.1. (i) For any positive integers j, k,   jk  X 1 X log p  c0 (j, k) 2ω(k) ϕ (k) = log (jk) + γ + + n k  p − 1 jk n=1 p|k

(n,k)=1

P

where ω (k) := p|k 1 and |c0 (j, k)| ≤ 1. (ii) For any positive integer k ≥ 9,

k ϕ (k)

2 ζ (2)

Y

1−

p|k

1 p2

2 X π log p  + 2γ1 + γ + − log k + ≤ 0. 3 p−1 

2

p|k

(iii) For any positive integer k ≥ 9, Y

1/ϕ(k)

|L (1; χ)|

χ6=χ0

1 Y p 1 2 ≤ ζ (2) . 1− 2 p p|k

Proof. (i) 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

where |ε (d)| ≤ 1 and hence: jk X µ (d) X µ (d) log d X 1 1 X = {log (jk) + γ} − + ε (d) µ (d) n d d jk n=1 d|k

(n,k)=1

d|k

and we conclude by noting that X µ (d) d|k

X µ (d) log d d|k


d

d

=

=−

d|k

ϕ (k) , k

ϕ (k) X log p k p−1 p|k


6


and X X ≤ ε (d) µ (d) µ2 (d) = 2ω(k) . d|k d|k (ii) Define A (k) :=

k ϕ (k)

2 ζ (2)

Y

1−

p|k

1 p2

2 X log p  π + 2γ1 + γ + − log k + . 3 p−1 2



p|k

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

and from the inequality ([4]) k 2.50637 < eγ log log k + ϕ (k) log log k valid for any integer k ≥ 3, we obtain 2 2 2 2.50637 π2 k γ A (k) ≤ ζ (2) e log log k + + 2γ1 + γ + − log k, then: −χ0 (p) Y 1 eγ ϕ (k) δ (x) 1− < · log x, p k p≤x where δ is the function defined in (1.2). Proof. Since x > 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) =

p≤x p-k

=

1 1− p

−1

1 1− p

−1 Y

Y

Y p≤x

p≤x p|k

1 1− p

−1 ϕ (k) Y 1 1− = k p≤x p and we use (1.1) .

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− := Π1 × Π2 p p p p≤x p≤x p≤x χ6=χ p≡l (mod k)


0


10


with Π1
x p α=2 p≤x and thus ( !) ∞ ∞ α Y X X X X X X χ (p) χ (p ) χ (p) + − Π2 = L (1; χ)χ(l) · exp χ (l) − α p αp αpα p α=2 p>x p≤x α=2 χ6=χ χ6=χ 0

0

and hence ) ∞ X X χ (p) XX 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>x p (

Y

χ6=χ0

χ6=χ0

and we use Lemma 4.1 (iii) and Lemma 5.1 (iii). We conclude the proof by noting that, if δ(x) x > 37, e2(γ−B) < 1. R EFERENCES [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. MITRINOVIC´ AND J. 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.

