A273556 Decimal expansion of Rosser's constant.

8, 3, 2, 4, 2, 9, 0, 6, 5, 6, 6, 1, 9, 4, 5, 2, 7, 8, 0, 3, 0, 8, 0, 5, 9, 4, 3, 5, 3, 1, 4, 6, 5, 5, 7, 5, 0, 4, 5, 4, 4, 5, 3, 1, 8, 0, 7, 7, 4, 1, 7, 0, 5, 3, 2, 4, 0, 8, 9, 3, 9, 9, 1, 2, 9, 6, 0, 3, 4, 7, 0, 7, 1, 3, 9, 4, 8, 1, 1, 4, 2, 4, 2, 1, 9, 1, 6, 2, 7, 2, 2, 5, 0, 4, 6, 3, 8, 1

Offset

0,1

Comments

Named after the American logician and mathematician John Barkley Rosser, Sr. (1907-1989). - Amiram Eldar, Jun 20 2021

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.1 Hardy-Littlewood constants, p. 86.

Links

Table of n, a(n) for n=0..97.

J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math., Vol. 6, No. 1 (1962), pp. 64-94, eq. (2.14).

Eric Weisstein's World of Mathematics, Twin Primes Constant.

Formula

4*C_2/exp(2*EulerGamma), where C_2 is the twin primes constant.

Equals lim_{x->inf} Product_{2 < p <= x} (1-2/p)*log(x)^2.

Example

0.832429065661945278030805943531465575045445318077417053240893991296...

Mathematica

digits = 98; s[n_] := (1/n)*N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], digits + 60]; C2 = (175/256)*Product[(Zeta[n]*(1 - 2^(-n))*(1 - 3^(-n) )*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[n]), {n, 2, digits + 60}];
RealDigits[4*C2/Exp[2*EulerGamma], 10, digits] // First

Prog

(PARI) 4 * exp(-2*Euler) * prodeulerrat(1-1/(p-1)^2, 1, 3) \\ Amiram Eldar, Mar 17 2021

Crossrefs

Cf. A005597, A091724, A246061.

Sequence in context: A198843 A346718 A119277 * A346713 A154014 A063568

Adjacent sequences: A273553 A273554 A273555 * A273557 A273558 A273559

Keyword

nonn,cons

Author

Jean-François Alcover, May 25 2016

Status

approved