This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333360 #57 Aug 24 2025 16:11:20
%S A333360 0,0,0,7,2,9,5,4,8,2,7,2,7,0,9,7,0,4,2,1,5,8,7,5,5,1,8,5,6,9,0,9,3,9,
%T A333360 7,0,5,0,3,3,5,1,5,0,5,7,0,3,5,5,4,2,3,7,3,5,8,9,6,5,2,7,4,4,6,6,6,1,
%U A333360 2,3,0,2,4,4,7,1,3,2,9,1,2,8,7,8,3,2,5,6,3,9,6,7,1,7,6,2,8,3,8,4,6,5,6,7,0,2,4,1,4,3,5,8,5,2,4
%N A333360 Decimal expansion of Sum_{n>=1} 1/z(n)^3 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.
%C A333360 a(1)-a(7) published by André Voros in 2001.
%C A333360 a(8)-a(20) computed by David Platt, Mar 15 2020.
%C A333360 a(21)-a(78) computed by _Fredrik Johansson_, Aug 04 2022 by mpmath procedure.
%C A333360 a(79)-a(350) computed by Artur Kawalec, Aug 15 2022 up to 350 decimal digits on basis algorithm of Juan Arias de Reyna.
%C A333360 a(351)-a(495) computed by Juan Arias de Reyna, using his implementation in mpmath from 2010, documented in his paper from 2020 (see link).
%C A333360 b-file on basis data from email Aug 16 2022 of Juan Arias de Reyna to _Artur Jasinski_.
%C A333360 Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
%C A333360 Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
%C A333360 Sum_{m>=1} 1/z(m)^3 = 0.0007295482727...; this sequence.
%C A333360 Sum_{m>=1} 1/z(m)^4 = 0.0000371725992...; see A335815.
%C A333360 Sum_{m>=1} 1/z(m)^5 = 0.0000022311886...; see A335814.
%C A333360 Sum_{m>=1} 1/z(m)^6 = 0.0000001441739...; see A335826.
%C A333360 Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.
%C A333360 Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.
%C A333360 Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.
%C A333360 Sum_{r>=1} Sum_{m>=n+1} 1/(z(r)*z(m))^3 = 0.00000619403... see A355283.
%H A333360 Artur Jasinski, <a href="/A333360/b333360.txt">Table of n, a(n) for n = 0..497</a>
%H A333360 mpmath, <a href="https://live.sympy.org/">Online versions</a>.
%H A333360 Artur Kawalec, <a href="https://arxiv.org/abs/2009.02640">The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function</a>, arxiv:2009.02640 [math.NT], 2020.
%H A333360 Artur Kawalec, <a href="https://arxiv.org/abs/2012.06581">Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function</a>, arxiv:2012.06581 [math.NT], 2021.
%H A333360 Artur Kawalec, <a href="https://arxiv.org/abs/2106.06915">The inverse Riemann zeta function</a>, arxiv:2106.06915 [math.NT], 2021, p. 38 formula (146).
%H A333360 Juan Arias de Reyna, <a href="https://arxiv.org/abs/2006.04869">Computation of the secondary zeta function</a>, arxiv:2006.04869 [math.NT], 2020.
%H A333360 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 A333360 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 A333360 André Voros, <a href="https://aif.centre-mersenne.org/item/AIF_2003__53_3_665_0/">Zeta functions for the Riemann zeros</a>, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, pp. 665-699.
%F A333360 No explicit formula is known (André Voros, personal communication to _Artur Jasinski_, Mar 09 2020).
%e A333360 0.00072954827270970421...
%o A333360 (Python)
%o A333360 from mpmath import *
%o A333360 mp.dps = 90
%o A333360 nprint(secondzeta(3), 78)
%Y A333360 Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645.
%K A333360 nonn,cons,changed
%O A333360 0,4
%A A333360 _Artur Jasinski_, Mar 16 2020