The Mertens constant ,
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 ,
(1)
where the sum is over primes and is a Landau symbol . This
sum is the analog of
(2)
where
is the Euler-Mascheroni constant (Gourdon
and Sebah).
The constant is given by the infinite sum
(3)
where
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
(4)
According to Lindqvist and Peetre (1997), this was shown independently by Meissel
in 1866 and Mertens (1874). Formula (3 ) is equivalent to
where
is the prime zeta function , which follows from
(5 ) using the Mercator series
for
with .
is also given by the rapidly converging
series
(7)
where
is the Riemann zeta function , and is the 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 ,
and Gourdon and Sebah give 100 digits.
The product of
behaves asymptotically as
(9)
(Hardy 1999, p. 57), where is the Euler-Mascheroni
constant and
is asymptotic notation , which is the Mertens
theorem .
The constant
also occurs in the 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 is the totient function
and
is the 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. Cambridge, MA: MIT Press,
1996. Ellison, W. J. and Ellison, F. Prime
Numbers. New York: Wiley, 1985. Finch, S. R. "Meissel-Mertens
Constants." §2.2 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 94-98,
2003. 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 Unpublished manuscript. 1996.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. New York:
Chelsea, 1999. Hardy, G. H. and Wright, E. M. "Mertens's
Theorem." §22.8 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Oxford University
Press, pp. 351-353 and 355, 1979. Ingham, A. E. The
Distribution of Prime Numbers. London: Cambridge University Press, pp. 22-24,
1990. Knuth, D. E. The
Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed.
Reading, MA: Addison-Wesley, 1998. Landau, E. Handbuch
der Lehre von der Verteilung der Primzahlen, 3rd ed. New York: Chelsea, pp. 100-102,
1974. Le Lionnais, F. Les
nombres remarquables. Paris: Hermann, p. 24, 1983. 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. New York: Springer-Verlag, 1971. 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. New York: Springer-Verlag,
1997. 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. Providence, RI: Amer. Math. Soc., p. 22,
2000. Titchmarsh, E. C. The
Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987. Referenced
on Wolfram|Alpha Mertens Constant
Cite this as:
Weisstein, Eric W. "Mertens Constant."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/MertensConstant.html
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