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

Author

Artur Jasinski, Jun 25 2020

Keywords

• nonn
• cons

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.

Examples

			0.0000371725992852696861648662624717405784536508897300...
		

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

Crossrefs

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

Programs

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]]]

Formula

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