The Mertens constant twin primes constant
and that appears in Mertens' second theorem ,
(1)
where the sum is over primes and Landau symbol . This
sum is the analog of
(2)
where Euler-Mascheroni constant (Gourdon
and Sebah).
The constant is given by the infinite sum
(3)
where Euler-Mascheroni constant and
(4)
According to Lindqvist and Peetre (1997), this was shown independently by Meissel
in 1866 and Mertens (1874). Formula (3 ) is equivalent to
where prime zeta function , which follows from
(5 ) using the Mercator series
for
(7)
where Riemann zeta function , and Möbius function
(Flajolet and Vardi 1996, Schroeder 1997, Knuth 1998).
The Mertens constant has the numerical value
(8)
(OEIS A077761 ). Knuth (1998) gives 40 digits of
The product of
(9)
(Hardy 1999, p. 57), where Euler-Mascheroni
constant and asymptotic notation , which is the Mertens
theorem .
The constant summatory function of the
number of distinct prime factors
(10)
(Hardy and Wright 1979, p. 355).
The related constant
(OEIS A083342 ) appears in the summatory function of the number of (not necessarily distinct) prime
factors
(15)
(Hardy and Wright 1979, p. 355), where totient function
and Riemann zeta function .
Another related constant is
(OEIS A083343 ; Rosser and Schoenfeld 1962, Montgomery 1971, Finch 2003), which appears in another equivalent form of the Mertens theorem
(18)
See also Brun's Constant ,
Harmonic Series ,
Mertens' Second Theorem ,
Prime Factor ,
Prime Number ,
Twin Primes Constant
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References Bach, E. and Shallit, J. Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Ellison, W. J. and Ellison, F. Prime
Numbers. Finch, S. R. "Meissel-Mertens
Constants." §2.2 in Mathematical
Constants. Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical
Constants." Unpublished manuscript. 1996. http://algo.inria.fr/flajolet/Publications/landau.ps . Gourdon,
X. and Sebah, P. "Some Constants from Number Theory." http://numbers.computation.free.fr/Constants/Miscellaneous/constantsNumTheory.html . Hardy,
G. H. Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. Hardy, G. H. and Wright, E. M. "Mertens's
Theorem." §22.8 in An
Introduction to the Theory of Numbers, 5th ed. Ingham, A. E. The
Distribution of Prime Numbers. Knuth, D. E. The
Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Landau, E. Handbuch
der Lehre von der Verteilung der Primzahlen, 3rd ed. Le Lionnais, F. Les
nombres remarquables. Lindqvist,
P. and Peetre, J. "On the Remainder in a Series of Mertens." Expos.
Math. 15 , 467-478, 1997. Mertens, F. J. für Math. 78 ,
46-62, 1874. Michon, G. P. "Final Answers: Numerical Constants."
http://home.att.net/~numericana/answer/constants.htm#mertens . Montgomery,
H. L. Topics
in Multiplicative Number Theory. Rosser,
J. B. and Schoenfeld, L. "Approximate Formulas for Some Functions of Prime
Numbers." Ill. J. Math. 6 , 64-94, 1962. Schroeder,
M. R. Number
Theory in Science and Communication, with Applications in Cryptography, Physics,
Digital Information, Computing, and Self-Similarity, 3rd ed. Sloane, N. J. A. Sequences A077761 ,
A083342 , and A083343
in "The On-Line Encyclopedia of Integer Sequences." Tenenbaum,
G. and Mendes-France, M. The
Prime Numbers and Their Distribution. Titchmarsh, E. C. The
Theory of the Riemann Zeta Function, 2nd ed. Referenced
on Wolfram|Alpha Mertens Constant
Cite this as:
Weisstein, Eric W. "Mertens Constant."
From MathWorld https://mathworld.wolfram.com/MertensConstant.html
Subject classifications More... Less...
,
also known as the Hadamard-de la Vallee-Poussin constant, prime reciprocal constant
(Bach and Shallit 1996, p. 234), or Kronecker's constant (Schroeder 1997), is
a constant related to the twin primes constant
and that appears in Mertens' second theorem,
is a Landau symbol. This
sum is the analog of
is the Euler-Mascheroni constant (Gourdon
and Sebah).
![B_1=gamma+sum_(k=1)^infty[ln(1-p_k^(-1))+1/(p_k)]](/images/equations/MertensConstant/NumberedEquation3.svg)
is the Euler-Mascheroni constant and
is the 
th prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979;
Le Lionnais 1983; Ellison and Ellison 1985), or by the limit
is the prime zeta function, which follows from
(5) using the Mercator series
for 
with 
.
is also given by the rapidly converging
series
![B_1=gamma+sum_(m=2)^infty(mu(m))/mln[zeta(m)],](/images/equations/MertensConstant/NumberedEquation5.svg)
is the Riemann zeta function, and 
is the Möbius function
(Flajolet and Vardi 1996, Schroeder 1997, Knuth 1998).
,
and Gourdon and Sebah give 100 digits.
behaves asymptotically as
is the Euler-Mascheroni
constant and 
is asymptotic notation, which is the Mertens
theorem.
also occurs in the summatory function of the
number of distinct prime factors 
,
![gamma+sum_(k=1)^(infty)[ln(1-p_k^(-1))+1/(p_k-1)]](/images/equations/MertensConstant/Inline27.svg)
![gamma+sum_(n=2)^(infty)(phi(n)ln[zeta(n)])/n](/images/equations/MertensConstant/Inline33.svg)
,
is the totient function
and 
is the Riemann zeta function.
