A335918 Decimal expansion of Sum_{m>=1} 1/(1/4 + z(m)^2)^2 where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function.

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

Offset: 0

Author

Artur Jasinski, Jun 29 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...; 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.

Examples

			0.0000371006364374648715125054339132797135962919799565287...
		

Links

• André Voros, Zeta functions for the Riemann zeros, arXiv:math/0104051 [math.CV], 2002-2003, p.22 Table 1.

Crossrefs

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

Programs

Mathematica

Join[{0, 0, 0, 0},RealDigits[N[3 + EulerGamma + EulerGamma^2 - Pi^2/8 - Log[4 Pi] + 2 StieltjesGamma[1], 105]][[1]]]

Formula

Equals: 3 + gamma + gamma^2 - Pi^2/8 - log(4*Pi) + 2*gamma(1), where gamma is the Euler-Mascheroni gamma constant (see A001620) and gamma(1) is 1st Stieltjes constant (see A082633).