A332614 a(n) is the smallest index k such that Sum_{m=1..k} 1/z(m) > n where z(m) is the imaginary part of the m-th nontrivial zero of the Riemann zeta function, n=0,1,2,...

1, 93, 621, 2437, 7438, 19490, 45996, 100462, 206617, 404855, 762155, 1387088, 2452209, 4227039, 7126088, 11778044, 19124514, 30559702, 48126380, 74788784, 114809974, 174270215, 261774713, 389414312, 574062463, 839117171, 1216829213, 1751399577, 2503082172, 3553595368

Offset: 0

Author

Artur Jasinski, Feb 17 2020

Keywords

• nonn

Comments

Because series Sum_{m>=1} 1/z(m) is divergent this sequence is infinite.

Sum_{m>=1} 1/z(m)^2 = 0.0231049931...; see A332645.

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.

a(11)-a(39) computed by David Platt, Mar 20 2020.

Examples

			a(0)=1 because 1/z(1) = 0.070747749954285585596 > 0
a(1)=93 because Sum_{m=1..93} 1/z(m) = 1.00082895080028509266 > 1
a(2)=621 because Sum_{m=1..621} 1/z(m) = 2.00017203211984838994 > 2.
		

Links

• Artur Jasinski, Table of n, a(n) for n = 0..39
• Kano Kono, Vieta's Formulas on Completed Riemann Zeta, Alien's Mathematics.

Crossrefs

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

Programs

Mathematica

aa = {}; kk = 0; b = 0; Do[b = b + N[1/Im[ZetaZero[n]], 30];
If[b > kk, AppendTo[aa, n]; kk = kk + 1];, {n, 1, 1000000}]; aa

Extensions

More terms from Artur Jasinski, Feb 21 2020