A084448 - OEIS

%I #30 Dec 17 2024 10:28:12

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

%T 4,2,7,6,4,0,3,6,3,8,0,3,3,5,2,0,1,7,8,3,6,6,6,5,2,2,3,0,6,3,5,7,3,5,

%U 9,6,9,9,6,6,6,5,7,7,1,7,2,7,5,9,5,2,5,1,0,0,3,3,2,5,0,8,7,5,5

%N Decimal expansion of (negative of) Kinkelin constant.

%C Named after the Swiss mathematician Hermann Kinkelin (1832-1913). - _Amiram Eldar_, Jun 16 2021

%H Gert Almkvist, <a href="https://projecteuclid.org/euclid.em/1047674152">Asymptotic formulas and generalized Dedekind sums</a>, Exper. Math., Vol. 7, No. 4 (1998), pp. 343-359.

%H Hermann Kinkelin, <a href="https://doi.org/10.1515/crll.1860.57.122">Ueber eine mit der Gammafunction verwandte Transcendente und deren Anwendung auf die Integralrechnung</a>, J. Reine Angew. Math., Vol. 57 (1860), pp. 122-158; <a href="https://eudml.org/doc/147780">alternative link</a>. See eq. (22), p. 133.

%H E. M. Wright, <a href="https://doi.org/10.1093/qmath/os-2.1.177">Asymptotic partition formulae, I: Plane partitions</a>, Quart. J. Math., Vol. 2 (1931), pp. 177-189.

%F Zeta(1, -1). Almkvist gives many formulas.

%F Equals (1 - gamma - log(2*Pi))/12 + Zeta'(2)/(2*Pi^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Jul 25 2015

%F From _Amiram Eldar_, Jun 16 2021: (Start)

%F Equals 1/24 - gamma/3 - Sum_{k>=1} (zeta(2*k+1)-1)/((2*k+1)*(2*k+3)) = 1/12 - log(A), where A is the Glaisher-Kinkelin constant (A074962) (Kinkelin, 1860).

%F Equals 2 * Integral_{x>=0} x*log(x)/(exp(2*Pi*x)-1) dx = 2*A261819. (Wright, 1931). (End)

%e -0.1654211437004509292139196602427806427640363803352017836665223...

%p Digits := 200; evalf(Zeta(1,-1));

%t RealDigits[1/12 - Log[Glaisher], 10, 99] // First (* _Jean-François Alcover_, Feb 15 2013 *)

%o (PARI) -zeta'(-1) \\ _Charles R Greathouse IV_, Dec 12 2013

%Y Cf. A001620, A074962, A084539.

%K nonn,cons,easy,changed

%O 0,2

%A _N. J. A. Sloane_, Jun 27 2003