login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2018 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A083343 Decimal expansion of constant B3 (or B_3) related to the Mertens constant. 9
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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
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 = int(x = 1, infty, (x - theta(x))/x^2) ... [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

Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, 2016, https://doi.org/10.1007/s11139-016-9839-4

S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 94-98.

Sh. T. Ishmukhametov, F. F. Sharifullina, On distribution of semiprime numbers, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2014, No. 8, pp. 53-59. English translation in Russian Mathematics, 2014, Volume 58, Issue 8 , pp 43-48

E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, 2nd ed., Chelsea, 1953, pp. 197-203.

LINKS

David Broadhurst, Table of n, a(n) for n = 1..300

David Broadhurst, The Mertens constant ...

David Broadhurst, 1000 digits

H. Cohen, High-precision calculation of Hardy-Littlewood constants, (1998).

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. Journ. Math. 6 (1962) 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)

Eric Weisstein's World of Mathematics, Mertens Constant

Don Zagier, Newman's short proof of the prime number theorem

FORMULA

B_3 = lim_{x to infty} ( log x - sum_{p <= x} log p / p ). - Dick Boland, Mar 09 2008

B_3 = EulerGamma - Sum_{n >= 2} P'(n), where P'(n) is the prime zeta P function derivative. - Jean-François Alcover, Apr 25 2016

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

See also A238114 = 1 + B_3.

Sequence in context: A214919 A290599 A070163 * A292527 A186111 A186813

Adjacent sequences:  A083340 A083341 A083342 * A083344 A083345 A083346

KEYWORD

nonn,cons

AUTHOR

Eric W. Weisstein, Apr 24 2003

EXTENSIONS

Edited by N. J. A. Sloane, Mar 05 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 11 19:37 EST 2018. Contains 318049 sequences. (Running on oeis4.)