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A332645 Decimal expansion of Sum_{n>=1} 1/z(n)^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

%I A332645 #63 Aug 06 2024 06:36:29
%S A332645 0,2,3,1,0,4,9,9,3,1,1,5,4,1,8,9,7,0,7,8,8,9,3,3,8,1,0,4,3,0,3,3,9,0,
%T A332645 1,4,0,0,3,3,8,1,7,6,0,3,9,7,4,2,2,0,9,0,1,2,3,1,8,2,5,0,0,5,6,0,7,6,
%U A332645 3,7,4,7,9,5,4,0,0,6,1,6,3,1,3,9,8,4,4,4,8,6,7,8,3,1,5,8,9,8,0,0,6,9,7,6,7,7
%N A332645 Decimal expansion of Sum_{n>=1} 1/z(n)^2 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.
%D A332645 J. P. Gram, "Note sur le calcul de la fonction zeta(s) de Riemann", Det Kgl. Danske Vid. Selsk. Overs., 1895, pp. 303-308. p.307 (16 decimal digits).
%D A332645 Charles Jean De La Vallée Poussin, Sur La Fonction de Riemann Et Le Nombre Des Nombres Premiers Inférieurs à Une Limite Donnee, 1899.
%H A332645 Jesús Guillera, <a href="https://arxiv.org/abs/1307.5723">Some sums over the non-trivial zeros of the Riemann zeta function</a>, arXiv:1307.5723v7 [math.NT], 2013-2014; see p.9 eq.(21).
%H A332645 Kano Kono, <a href="http://fractional-calculus.com/vieta%27s_formulas_completed_riemann_zeta.pdf">Vieta's Formulas on Completed Riemann Zeta</a>, Alien's Mathematics p. 13 eq. 2.3(2).
%H A332645 E. Landau, <a href="https://archive.org/details/handbuchderlehre01landuoft/page/314/mode/2up"> Handbuch der Lehre von der Verteilung der Primzahlen I</a>, 1909, pp. 310-342.
%H A332645 J. J. Y. Liang and John Todd, <a href="https://archive.org/details/jresv76Bn3-4p161/page/n15/mode/2up">The Stieltjes Constants</a>, Journal of Research of the National Bureau of Standards, 1972, pp. 175-176.
%H A332645 André Voros, <a href="https://arxiv.org/abs/math/0104051">Zeta functions for the Riemann zeros</a>, arXiv:math/0104051 [math.CV], 2002-2003, p.25 Table 2.
%H A332645 André Voros, <a href="https://citeseerx.ist.psu.edu/pdf/937e9d9c737304007c6af6a58901a8c1e530df31">Zeta functions for the Riemann zeros</a>, 2001(2008) p.20 Table 1.
%H A332645 André Voros, <a href="https://doi.org/10.5802/aif.1955">Zeta functions for the Riemann zeros</a>, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, p. 665-699.
%H A332645 André Voros, <a href="https://www.springer.com/gp/book/9783642052026">Zeta functions over Zeros of the Zeta functions</a>, 2010, p. 153.
%F A332645 Equals -4 + G + Pi^2/8 + (1/2)(zeta''(1/2)/zeta(1/2) - (zeta'(1/2)/zeta(1/2))^2) where G is the Catalan constant A006752.
%F A332645 Equals G - 4 + (Pi^2 - (gamma + Pi/2 + log(8*Pi))^2) / 8 + zeta''(1/2) / (2*zeta(1/2)), where gamma is the Euler-Mascheroni constant A001620 and G is the Catalan constant A006752. - _Vaclav Kotesovec_, Feb 19 2020
%F A332645 Also equals (-32 - log(Pi)^2 + psi(0, 1/4)^2 + psi(1, 1/4) + 4*(psi(0, 1/4) * zeta'(1/2) + zeta''(1/2)) / zeta(1/2)) / 8, where psi(0, 1/4) = -A020777 and psi(1, 1/4) = A282823. - _Vaclav Kotesovec_, Feb 19 2020
%e A332645 0.0231049931154189707889338104303390140033817603974220901231825...
%p A332645 evalf((-32 - log(Pi)^2 + Psi(0, 1/4)^2 + Psi(1, 1/4) + 4*(Psi(0, 1/4) * Zeta(1, 1/2) + Zeta(2, 1/2)) / Zeta(1/2)) / 8, 120); # _Vaclav Kotesovec_, Feb 19 2020
%t A332645 Join[{0}, RealDigits[N[-4 + Catalan + Pi^2/8 + (Zeta''[1/2]/Zeta[1/2] - (Zeta'[1/2] / Zeta[1/2])^2)/2, 105]][[1]]]
%t A332645 N[SeriesCoefficient[Log[s*(s-1)*Pi^(-s/2)*Gamma[s/2]*Zeta[s]/2], {s, 1/2, 2}], 105] (* _Vaclav Kotesovec_, Feb 19 2020 *)
%Y A332645 Cf. A006752, A074760, A332614.
%K A332645 nonn,cons
%O A332645 0,2
%A A332645 _Artur Jasinski_, Feb 18 2020