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

0,2

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).

Links

Table of n, a(n) for n=0..86.

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.

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.

Example

0.14248676756297667766013119038516485825699065019561...

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]]

Crossrefs

Cf. A001620, A002117, A006752, A077761.

Cf. A001008, A002805.

Sequence in context: A154973 A154793 A016693 * A137718 A007361 A128136

Adjacent sequences: A345410 A345411 A345412 * A345414 A345415 A345416

Keyword

nonn,cons

Author

Amiram Eldar, Jun 18 2021

Status

approved