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,


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


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

The constant is given by the infinite sum


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


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


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


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


(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


(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),


(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 Omega(n),


(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


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


See also

Brun's Constant, Harmonic Series, Mertens' Second Theorem, Prime Factor, Prime Number, Twin Primes Constant

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Mertens Constant

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Weisstein, Eric W. "Mertens Constant." From MathWorld--A Wolfram Web Resource.

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