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

0,5

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...; 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

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.22 Table 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).

Example

0.0000371006364374648715125054339132797135962919799565287...

Mathematica

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

Crossrefs

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

Sequence in context: A247956 A295043 A074051 * A048292 A072450 A085785

Adjacent sequences: A335915 A335916 A335917 * A335919 A335920 A335921

Keyword

nonn,cons

Author

Artur Jasinski, Jun 29 2020

Status

approved