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Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))*(log(z(m)/(2*Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.
2

%I #117 Oct 08 2022 00:00:26

%S 1,2,4,9,13,27,34,135,159,186,212,315,363,453,693,922,1496,4765,6709,

%T 44555,73997,82552,87761,95248,415587,420891,1115578,8546950,24360732,

%U 41820581,1048449114,3570918901,35016977796

%N Numbers m such that d(m) < d(k) for all k < m, where d is the normalized delta defined as d(m) = (z(m+1) - z(m))*(log(z(m)/(2*Pi))/(2*Pi)) where z(k) is the imaginary part of the k-th Riemann zeta zero.

%C No more records up to k = 103800788359.

%C Indices of zeros for successive maximal records of the normalized delta see A329742.

%C a(28)-a(33) computed by David Platt (2020).

%C Conjectural next terms: 1217992279429, 4088664936219.

%H Xavier Gourdon, <a href="https://www.semanticscholar.org/paper/The-10-13-first-zeros-of-the-Riemann-Zeta-function-Xavier/6eff62ff5d98e8ad2ad8757c0faf4bac87546f27">The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height</a>, 2004.

%H David Platt, <a href="/A328656/a328656.txt">Results computation of the smallest relative gaps between successive zeta zeros, 2020.</a>

%e n | a(n) | d(n)

%e ---+-------+------------

%e 1 | 1 | 0.88871193

%e 2 | 2 | 0.76669277

%e 3 | 4 | 0.63017799

%e 4 | 9 | 0.57239954

%e 5 | 13 | 0.53062398

%e 6 | 27 | 0.52634271

%e 7 | 34 | 0.38628922

%e 8 | 135 | 0.37238098

%e 9 | 159 | 0.35780768

%e 10 | 186 | 0.32438582

%e 11 | 212 | 0.29105188

%e 12 | 315 | 0.24707528

%e 13 | 363 | 0.24343744

%e 14 | 453 | 0.23631515

%e 15 | 693 | 0.18028720

%e 16 | 922 | 0.13762601

%e 17 | 1496 | 0.08925253

%e 18 | 4765 | 0.04628960

%e 19 | 6709 | 0.04209838

%e 20 | 44555 | 0.04074628

%t prec = 30; min = 10; aa = {}; Do[kk = N[Im[(ZetaZero[n + 1] - ZetaZero[n])], prec] (Log[N[Im[ZetaZero[n]], prec]/(2 Pi)]/(2 Pi));

%t If[kk <min, min = kk; AppendTo[aa, n]], {n, 1, 2000000}]; aa

%Y Cf. A114856, A254297, A255739, A255742, A326502, A329742.

%K nonn

%O 1,2

%A _Artur Jasinski_, Jan 03 2020