A335814 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.

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, 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, 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

Offset

0,6

Comments

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

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

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

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

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

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

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

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

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

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

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.

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

Links

Artur Jasinski, Table of n, a(n) for n = 0..350

Artur Kawalec, The recurrence formulas for primes and non-trivial zeros of the Riemann zeta function, arxiv:2009.02640 [math.NT], 2020.

Artur Kawalec, Analytical recurrence formulas for non-trivial zeros of the Riemann zeta function, arxiv:2012.06581 [math.NT], 2021.

Artur Kawalec, The inverse Riemann zeta function, arxiv:2106.06915 [math.NT], 2021 p. 38 formula (146).

Juan Arias de Reyna, Computation of the secondary zeta function, arxiv:2006.04869 [math.NT], 2020.

Formula

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).

Example

0.0000022311886995021033286406286918...

Prog

(Python)
from mpmath import *
mp.dps = 90
nprint(secondzeta(5), 78)

Crossrefs

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

Sequence in context: A264010 A071435 A365484 * A119428 A241815 A051521

Adjacent sequences: A335811 A335812 A335813 * A335815 A335816 A335817

Keyword

nonn,cons

Author

Artur Jasinski, Jun 25 2020

Status

approved