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A335826
Decimal expansion of Sum_{n>=1} 1/z(n)^6 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.
5
0, 0, 0, 0, 0, 0, 1, 4, 4, 1, 7, 3, 9, 3, 1, 4, 0, 0, 9, 7, 3, 2, 7, 9, 6, 9, 5, 3, 8, 1, 5, 5, 6, 0, 9, 4, 8, 2, 0, 9, 0, 7, 0, 3, 6, 8, 8, 3, 0, 0, 8, 5, 0, 9, 0, 9, 8, 1, 1, 8, 7, 1, 5, 9, 9, 9, 3, 6, 4, 2, 1, 7, 9, 0, 5, 3, 9, 4, 6, 3, 1, 6, 8, 9, 6, 4, 0, 8, 1, 9, 5, 5, 0, 6, 7, 4, 2, 0, 4, 6, 8, 3, 8, 8, 8, 3, 4, 2, 3, 0, 5
OFFSET
0,8
COMMENTS
Sum_{m>=1} 1/z(m) is a divergent series; see A332614.
Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.
Sum_{m>=1} 1/z(m)^3 = 0.0007295482727097...; see A333360.
Sum_{m>=1} 1/z(m)^4 = 0.0000371725992852...; see A335815.
Sum_{m>=1} 1/z(m)^5 = 0.0000022311886995...; see A335814.
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.
LINKS
André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.25 Table 2.
André Voros, Zeta functions for the Riemann zeros, 2001(2008) p.20 Table 1.
André Voros, Zeta functions for the Riemann zeros, Annales de l'Institut Fourier, Tome 53 (2003) no. 3, p. 665-699.
FORMULA
Universal formula for Sum_{n>=1} 1/z(n)^(2m) published in Voros 2002-2003 p. 22 (see Mathematica procedure below).
EXAMPLE
0.000000144173931400973279695381556....
MATHEMATICA
m = 3; Join[{0, 0, 0, 0, 0, 0}, RealDigits[N[((-1)^m (2^(2 m) - ((2^(2 m) - 1) Zeta[2 m] + (Zeta[2 m, 1/4] - Zeta[2 m, 3/4])/2^(2 m))/4 - (D[Log[Zeta[x]], {x, 2 m}] /. x -> 1/2)/(2 (2 m - 1)!) )), 105]][[1]]]
KEYWORD
nonn,cons
AUTHOR
Artur Jasinski, Jun 25 2020
STATUS
approved