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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.
9
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 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)
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jun 25 2020
STATUS
approved