%I
%S 1,3,3,2,5,8,2,2,7,5,7,3,3,2,2,0,8,8,1,7,6,5,8,2,8,7,7,6,0,7,1,0,2,7,
%T 7,4,8,8,3,8,4,5,9,4,8,9,0,4,2,4,2,2,6,6,1,7,8,7,1,3,0,8,9,9,7,5,7,3,
%U 4,0,0,4,1,7,1,9,3,0,4,0,1,8,6,8,7,5,4,8,0,4,5,5,1,4,1,6,8,6,2
%N Decimal expansion of constant B3 (or B_3) related to the Mertens constant.
%C Comment from _David Broadhurst_ Feb 26 2014, via a posting to the Number Theory Mailing List, and included here with permission. This comment concerns the number C = 1 + B_3 = 2.33258227573322088... (see also A238114). (Start)
%C In a beautifully concise and clear paper, "Newman's short proof of the prime number theorem", Don Zagier condensed work by Euler, Riemann, Chebyshev, de la Vallee Poussin, Hadamard, Mertens and, more recently, D. J. Newman (in 1980), to achieve a short selfcontained proof of the prime number theorem that is within the reach of a reader who understands enough real analysis to apply the meanvalue theorem and enough complex analysis to apply Cauchy's theorem.
%C The heart of this proof is the convergence of the integral
%C C = int(x = 1, infty, (x  theta(x))/x^2) ... [1]
%C where theta(x) = sum(prime p <= x, log(p)) is the sum of the natural logs of all the primes not exceeding x. Then the Prime Number Theorem follows in the form theta(x) ~ x, for otherwise the integral in [1] would not converge.
%C In a sense, this constant C is rather significant: if it did not exist the proof would fail. However, its actual value is a matter of sublime indifference to a true mathematician. To prove that it exists, one may use the equivalent expression
%C C = 1 + Euler + sum(prime p, log(p)/(p^2p)) ... [2]
%C that follows from Zagier's account. Here the sum is over all the positive primes and clearly converges, since the corresponding sum over integers n > 1 converges.
%C It is also easy, if unnecessary, to show that
%C C = 1 + Euler + sum(s > 1, mu(s)*zeta'(s)/zeta(s)) ... [3]
%C where mu(s) is the Moebius function. An approximate evaluation of this formula requires the derivatives zeta'(s) of Riemann's zeta(s) = sum(n > 0, 1/n^s) at sufficiently many squarefree integers s > 1.
%C By use of both [2] and [3], J. Barkley Rosser and Lowell Schoenfeld obtained (effectively) 16 good digits of C in "Approximate formulas for some functions of prime numbers", where they gave, in (2.11), a numerical result for 1  C.
%C A better way to compute C, however, is by use of a method indicated in Henri Cohen's paper "High precision computation of HardyLittlewood constants".
%C (End)
%D Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, 2016, https://doi.org/10.1007/s1113901698394
%D S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 9498.
%D Sh. T. Ishmukhametov, F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 5359. English translation in Russian Mathematics, 2014, Volume 58, Issue 8 , pp 4348
%D E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2nd ed., Chelsea, 1953, pp. 197203.
%H David Broadhurst, <a href="/A083343/b083343.txt">Table of n, a(n) for n = 1..300</a>
%H David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/cohenb3.ps">The Mertens constant ...</a>
%H David Broadhurst, <a href="http://physics.open.ac.uk/~dbroadhu/cert/cohenb3.txt">1000 digits</a>
%H H. Cohen, <a href="http://www.math.ubordeaux.fr/~cohen/hardylw.dvi">Highprecision calculation of HardyLittlewood constants</a>, (1998).
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="http://projecteuclid.org/euclid.ijm/1255631807">Approximate formulas for some functions of prime numbers</a>, Ill. Journ. Math. 6 (1962) 6494.
%H J. Barkley Rosser and Lowell Schoenfeld, <a href="/A000720/a000720.html">Approximate formulas for some functions of prime numbers</a> (scan of some key pages from an ancient annotated photocopy)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MertensConstant.html">Mertens Constant</a>
%H Don Zagier, <a href="http://people.mpimbonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf">Newman's short proof of the prime number theorem</a>
%F B_3 = lim_{x to infty} ( log x  sum_{p <= x} log p / p ).  Dick Boland, Mar 09 2008
%F B_3 = EulerGamma  Sum_{n >= 2} P'(n), where P'(n) is the prime zeta P function derivative.  _JeanFrançois Alcover_, Apr 25 2016
%e 1.3325822757332208817658287760710277488384594890424226617871308997573400417193...
%t digits = 99; B3 = EulerGamma  NSum[PrimeZetaP'[n], {n, 2, 10^4}, WorkingPrecision > 2 digits, NSumTerms > 200]; RealDigits[B3, 10, digits][[1]] (* _JeanFrançois Alcover_, Apr 25 2016 *)
%Y See also A238114 = 1 + B_3.
%K nonn,cons
%O 1,2
%A _Eric W. Weisstein_, Apr 24 2003
%E Edited by _N. J. A. Sloane_, Mar 05 2014
