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Mertens Constant


The Mertens constant B_1, 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,

 sum_(p<=x)1/p=lnlnx+B_1+o(1),
(1)

where the sum is over primes and o(1) is a Landau symbol. This sum is the analog of

 sum_(n<=x)1/n=lnx+gamma+o(1),
(2)

where gamma is the Euler-Mascheroni constant (Gourdon and Sebah).

The constant is given by the infinite sum

 B_1=gamma+sum_(k=1)^infty[ln(1-p_k^(-1))+1/(p_k)]
(3)

where gamma is the Euler-Mascheroni constant and p_k is the kth prime (Rosser and Schoenfeld 1962; Hardy and Wright 1979; Le Lionnais 1983; Ellison and Ellison 1985), or by the limit

 B_1=lim_(x->infty)(sum_(p<=x)1/p-lnlnx).
(4)

According to Lindqvist and Peetre (1997), this was shown independently by Meissel in 1866 and Mertens (1874). Formula (3) is equivalent to

B_1=gamma-sum_(k=1)^(infty)sum_(j=2)^(infty)1/(jp_k^j),
(5)
=gamma-sum_(j=2)^(infty)(P(j))/j,
(6)

where P(n) is the prime zeta function, which follows from (5) using the Mercator series for ln(1+x) with x=-1/p_k. B_1 is also given by the rapidly converging series

 B_1=gamma+sum_(m=2)^infty(mu(m))/mln[zeta(m)],
(7)

where zeta(n) is the Riemann zeta function, and mu(n) is the Möbius function (Flajolet and Vardi 1996, Schroeder 1997, Knuth 1998).

The Mertens constant has the numerical value

 B_1=0.2614972128...
(8)

(OEIS A077761). Knuth (1998) gives 40 digits of B_1, and Gourdon and Sebah give 100 digits.

The product of 1-1/p behaves asymptotically as

 product_(p<=x)(1-1/p)∼(e^(-gamma))/(lnx)
(9)

(Hardy 1999, p. 57), where gamma is the Euler-Mascheroni constant and ∼ is asymptotic notation, which is the Mertens theorem.

The constant B_1 also occurs in the summatory function of the number of distinct prime factors omega(k),

 sum_(k=2)^nomega(k)=nlnlnn+B_1n+o(n)
(10)

(Hardy and Wright 1979, p. 355).

The related constant

B_2=gamma+sum_(k=1)^(infty)[ln(1-p_k^(-1))+1/(p_k-1)]
(11)
=B_1+sum_(k=1)^(infty)1/(p_k(p_k-1))
(12)
=gamma+sum_(n=2)^(infty)(phi(n)ln[zeta(n)])/n
(13)
=1.034653...
(14)

(OEIS A083342) appears in the summatory function of the number of (not necessarily distinct) prime factors Omega(n),

 sum_(n<=x)Omega(n)=xlnlnx+B_2x+o(x)
(15)

(Hardy and Wright 1979, p. 355), where phi(n) is the totient function and zeta(n) is the Riemann zeta function.

Another related constant is

B_3=gamma+sum_(j=2)^(infty)sum_(k=1)^(infty)(lnp_k)/(p_k^j)
(16)
=1.3325822757...
(17)

(OEIS A083343; Rosser and Schoenfeld 1962, Montgomery 1971, Finch 2003), which appears in another equivalent form of the Mertens theorem

 B_3=lim_(x->infty)(lnx-sum_(p<=x)(lnp)/p).
(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.

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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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