A335815 Decimal expansion of Sum_{n>=1} 1/z(n)^4 where z(n) is the imaginary part of the n-th nontrivial zero of the Riemann zeta function.

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

Offset

0,5

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...; this sequence.

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

Table of n, a(n) for n=0..108.

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.

André Voros, Zeta functions over Zeros of the Zeta functions, 2010, p. 153.

Formula

Equals 16-Pi^4/24+(Zeta[4,3/4]-Zeta[4,1/4])/64-(Log[Zeta[x]]''''[1/2])/24

Example

0.0000371725992852696861648662624717405784536508897300...

Mathematica

Join[{0, 0, 0, 0}, RealDigits[N[-1/12*(D[Log[Zeta[x]], {x, 4}]/. x -> 1/2) - 1/24 Pi^4 -(Zeta[4, 1/4] - Zeta[4, 3/4])/64 + 16, 105]][[1]]]

Crossrefs

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

Sequence in context: A354977 A080172 A133065 * A021733 A021273 A097263

Adjacent sequences: A335812 A335813 A335814 * A335816 A335817 A335818

Keyword

nonn,cons

Author

Artur Jasinski, Jun 25 2020

Status

approved