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 A335826 Decimal expansion of Sum_{n>=1} 1/z(n)^6 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function. 2
 0, 0, 0, 0, 0, 0, 1, 4, 4, 1, 7, 3, 9, 3, 1, 4, 0, 0, 9, 7, 3, 2, 7, 9, 6, 9, 5, 3, 8, 1, 5, 5, 6, 0, 9, 4, 8, 2, 0, 9, 0, 7, 0, 3, 6, 8, 8, 3, 0, 0, 8, 5, 0, 9, 0, 9, 8, 1, 1, 8, 7, 1, 5, 9, 9, 9, 3, 6, 4, 2, 1, 7, 9, 0, 5, 3, 9, 4, 6, 3, 1, 6, 8, 9, 6, 4, 0, 8, 1, 9, 5, 5, 0, 6, 7, 4, 2, 0, 4, 6, 8, 3, 8, 8, 8, 3, 4, 2, 3, 0, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Sum_{m>=1} 1/z(m) is a divergent series; see A332614. Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645. Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360. Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815. Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814. Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760. Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275. Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276. LINKS 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. FORMULA Universal formula for Sum_{n>=1} 1/z(n)^(2m) published in Voros 2002-2003 p. 22 (see Mathematica procedure below). EXAMPLE 0.000000144173931400973279695381556.... MATHEMATICA m = 3; Join[{0, 0, 0, 0, 0, 0}, RealDigits[N[((-1)^m (2^(2 m) - ((2^(2 m) - 1) Zeta[2 m] + (Zeta[2 m, 1/4] - Zeta[2 m, 3/4])/2^(2 m))/4 - (D[Log[Zeta[x]], {x, 2 m}] /. x -> 1/2)/(2 (2 m - 1)!) )), 105]][[1]]] CROSSREFS Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815. Sequence in context: A021878 A247252 A016495 * A337191 A341863 A047213 Adjacent sequences:  A335823 A335824 A335825 * A335827 A335828 A335829 KEYWORD nonn AUTHOR Artur Jasinski, Jun 25 2020 STATUS approved

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Last modified September 19 10:53 EDT 2021. Contains 347556 sequences. (Running on oeis4.)