A332614 - OEIS

%I #58 Mar 30 2020 10:29:24

%S 1,93,621,2437,7438,19490,45996,100462,206617,404855,762155,1387088,

%T 2452209,4227039,7126088,11778044,19124514,30559702,48126380,74788784,

%U 114809974,174270215,261774713,389414312,574062463,839117171,1216829213,1751399577,2503082172,3553595368

%N 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,...

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

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

%C Sum_{m>=1} 1/(1/4 + z(m)^2) = 0.023095708966... see A074760.

%C Sum_{m>=1} 1/(1/2 + i*z(m))^2 + 1/(1/2 - i*z(m))^2 = -0.046154317... see A245275.

%C Sum_{m>=1} 1/(1/2 + i*z(m))^3 + 1/(1/2 - i*z(m))^3 = -0.00011115823... see A245276.

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

%H Artur Jasinski, <a href="/A332614/b332614.txt">Table of n, a(n) for n = 0..39</a>

%H Kano Kono, <a href="http://fractional-calculus.com/vieta%27s_formulas_completed_riemann_zeta.pdf">Vieta's Formulas on Completed Riemann Zeta</a>, Alien's Mathematics.

%e a(0)=1 because 1/z(1) = 0.070747749954285585596 > 0

%e a(1)=93 because Sum_{m=1..93} 1/z(m) = 1.00082895080028509266 > 1

%e a(2)=621 because Sum_{m=1..621} 1/z(m) = 2.00017203211984838994 > 2.

%t aa = {}; kk = 0; b = 0; Do[b = b + N[1/Im[ZetaZero[n]], 30];

%t If[b > kk, AppendTo[aa, n]; kk = kk + 1];, {n, 1, 1000000}]; aa

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

%K nonn

%O 0,2

%A _Artur Jasinski_, Feb 17 2020

%E More terms from _Artur Jasinski_, Feb 21 2020