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.

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, 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, 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

Offset: 0

Author

Artur Jasinski, Feb 18 2020

Keywords

• nonn
• cons

Examples

			0.0231049931154189707889338104303390140033817603974220901231825...
		

References

• 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).
• Charles Jean De La Vallée Poussin, Sur La Fonction de Riemann Et Le Nombre Des Nombres Premiers Inférieurs à Une Limite Donnee, 1899.

Links

• Jesús Guillera, Some sums over the non-trivial zeros of the Riemann zeta function, arXiv:1307.5723v7 [math.NT], 2013-2014; see p.9 eq.(21).
• Kano Kono, Vieta's Formulas on Completed Riemann Zeta, Alien's Mathematics p. 13 eq. 2.3(2).
• E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen I, 1909, pp. 310-342.
• J. J. Y. Liang and John Todd, The Stieltjes Constants, Journal of Research of the National Bureau of Standards, 1972, pp. 175-176.
• André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.25 Table 2.
• André Voros, Zeta functions for the Riemann zeros, 2001(2008) p.20 Table 1.
• André Voros, Zeta functions for the Riemann zeros, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, p. 665-699.
• André Voros, Zeta functions over Zeros of the Zeta functions, 2010, p. 153.

Crossrefs

Cf. A006752, A074760, A332614.

Programs

Maple

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

Mathematica

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]]]
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 *)

Formula

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.

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

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