OFFSET
1,2
COMMENTS
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)
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 self-contained proof of the prime number theorem that is within the reach of a reader who understands enough real analysis to apply the mean-value theorem and enough complex analysis to apply Cauchy's theorem.
The heart of this proof is the convergence of the integral
C = Integral_{x = 1..oo} (x - theta(x))/x^2 dx ... [1]
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.
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 = 1 + Euler + Sum_{prime p} log(p)/(p^2-p) ... [2]
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.
It is also easy, if unnecessary, to show that
C = 1 + Euler + Sum_{s > 1} mu(s)*zeta'(s)/zeta(s) ... [3]
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.
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.
A better way to compute C, however, is by use of a method indicated in Henri Cohen's paper "High precision computation of Hardy-Littlewood constants".
(End)
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.
Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2nd ed., Chelsea, 1953, pp. 197-203.
József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter VI, p. 199.
LINKS
David Broadhurst, Table of n, a(n) for n = 1..300
David Broadhurst, The Mertens constant, 2005.
David Broadhurst, 1000 digits.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, Vol. 45 (2018), pp. 227-251.
Pierre Dusart, On the divergence of the sum of prime reciprocals, WSEAS Transactions on Math. (2023) Vol.22, 508-513.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 196.
Sh. T. Ishmukhametov and F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59; English translation, Russian Mathematics, Vol. 58, No. 8 (2014), pp. 43-48.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. Journ. Math., Vol. 6 (1962), pp. 64-94.
J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers (scan of some key pages from an ancient annotated photocopy).
D. R. Ward, Some Series Involving Euler's Function, Journal of the London Mathematical Society, Vol. 1, No. 4 (1927), pp. 210-214.
Eric Weisstein's World of Mathematics, Mertens Constant.
Don Zagier, Newman's short proof of the prime number theorem, The American Mathematical Monthly, Vol. 104, No. 8 (1997), pp. 705-708.
FORMULA
Equals lim_{x->oo} (log x - Sum_{p <= x} log(p)/p). - Dick Boland, Mar 09 2008
Equals EulerGamma - Sum_{n >= 2} P'(n), where P'(n) is the prime zeta P function derivative. - Jean-François Alcover, Apr 25 2016
Equals lim_{n->oo} Sum_{k=1..n} mu(k)^2/phi(k) - log(n) (Ward, 1927). - Amiram Eldar, Mar 05 2021
EXAMPLE
1.3325822757332208817658287760710277488384594890424226617871308997573400417193...
MATHEMATICA
digits = 99; B3 = EulerGamma - NSum[PrimeZetaP'[n], {n, 2, 10^4}, WorkingPrecision -> 2 digits, NSumTerms -> 200]; RealDigits[B3, 10, digits][[1]] (* Jean-François Alcover, Apr 25 2016 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Eric W. Weisstein, Apr 24 2003
EXTENSIONS
Edited by N. J. A. Sloane, Mar 05 2014
STATUS
approved