A335814 - OEIS

%I #48 Aug 28 2022 16:51:23

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

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

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

%N Decimal expansion of Sum_{n>=1} 1/z(n)^5 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

%C a(1)-a(34) computed by David Platt, Mar 15 2020.

%C a(35)-a(78) computed by Fredrik Johansson, Aug 04 2022 by mpmath procedure.

%C a(79)-a(115) computed by Artur Kawalec, Aug 15 2022 up to 350 decimal digits on basis Juan Arias de Reyna algorithm.

%C b-file on basis data from email Aug 15 2022 from Artur Kawalec to Artur Jasinski.

%C Sum_{m>=1} 1/z(m) is a divergent series; see A332614.

%C Sum_{m>=1} 1/z(m)^2 = 0.0231049931154...; see A332645.

%C Sum_{m>=1} 1/z(m)^3 = 0.0007295482727...; see A333360.

%C Sum_{m>=1} 1/z(m)^4 = 0.0000371725992...; see A335815.

%C Sum_{m>=1} 1/z(m)^5 = 0.0000022311886...; see A335814.

%C Sum_{m>=1} 1/z(m)^6 = 0.0000001441739...; see A335826.

%C Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966...; see A074760.

%C Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317...; see A245275.

%C Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823...; see A245276.

%C Sum_{r>=1} Sum_{m>=n+1} 1/(z(r)*z(m))^3 = 0.00000619403... see A355283.

%H Artur Jasinski, <a href="/A335814/b335814.txt">Table of n, a(n) for n = 0..350</a>

%H 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 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 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 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.

%F No explicit formula for Sum_{n>=1} 1/z(n)^k is known for odd exponents k (Andre Voros, personal communication to Artur Jasinski, Mar 09 2020).

%e 0.0000022311886995021033286406286918...

%o (Python)

%o from mpmath import *

%o mp.dps = 90

%o nprint(secondzeta(5), 78)

%Y Cf. A013629, A074760, A104539, A104540, A104541, A104542, A245275, A245276, A306339, A306340, A306341, A332645, A333360, A335814, A335815, A335826, A355283.

%K nonn,cons

%O 0,6

%A _Artur Jasinski_, Jun 25 2020