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

Author

Artur Jasinski, Jun 25 2020

Keywords

• nonn
• cons
• changed

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.

Examples

			0.0000022311886995021033286406286918...
		

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.

Crossrefs

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

Programs

Python

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

Formula

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