A345413 Decimal expansion of exp(gamma + M)*(G - 7*zeta(3)/(4*Pi))/4, where gamma is Euler's constant (A001620), M is Mertens's constant (A077761) and G is Catalan's constant (A006752).

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

Offset: 0

Author

Amiram Eldar, Jun 18 2021

Keywords

• nonn
• cons

Comments

This constant is notable for being the asymptotic limit in a formula derived by Sinha and Wolf (2010) which "brings together the elements from nine different topics of number theory" (see the Formula section).

Examples

			0.14248676756297667766013119038516485825699065019561...
		

Links

• Nilotpal Kanti Sinha and Marek Wolf, On a unified theory of numbers, arXiv:1009.4810 [math.NT], 2010-2011. See section 8, p. 11, eq. 37.

Crossrefs

Cf. A001620, A002117, A006752, A077761.

Cf. A001008, A002805.

Programs

Mathematica

M = EulerGamma - NSum[PrimeZetaP[k]/k, {k, 2, Infinity}, WorkingPrecision -> 300, NSumTerms -> 300]; RealDigits[Exp[EulerGamma + M]*(Catalan - 7*Zeta[3]/(4*Pi))/4, 10, 100][[1]]

Formula

Equals lim_{n->oo} (1/log(n)^2) * Sum_{k=1..n} (1/gamma_k) * (1/k + 1/prime(k)) * (arctan(gamma_k/gamma_n))^2 * exp(H(k) + Sum_{i=1..k} 1/prime(i))), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and gamma_k is the imaginary part of the k-th nontrivial zero of the Riemann zeta function.