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A328656 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.

%I A328656 #117 Oct 08 2022 00:00:26
%S A328656 1,2,4,9,13,27,34,135,159,186,212,315,363,453,693,922,1496,4765,6709,
%T A328656 44555,73997,82552,87761,95248,415587,420891,1115578,8546950,24360732,
%U A328656 41820581,1048449114,3570918901,35016977796
%N A328656 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 A328656 No more records up to k = 103800788359.
%C A328656 Indices of zeros for successive maximal records of the normalized delta see A329742.
%C A328656 a(28)-a(33) computed by David Platt (2020).
%C A328656 Conjectural next terms: 1217992279429, 4088664936219.
%H A328656 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 A328656 David Platt, <a href="/A328656/a328656.txt">Results computation of the smallest relative gaps between successive zeta zeros, 2020.</a>
%e A328656    n |  a(n) |    d(n)
%e A328656   ---+-------+------------
%e A328656    1 |     1 | 0.88871193
%e A328656    2 |     2 | 0.76669277
%e A328656    3 |     4 | 0.63017799
%e A328656    4 |     9 | 0.57239954
%e A328656    5 |    13 | 0.53062398
%e A328656    6 |    27 | 0.52634271
%e A328656    7 |    34 | 0.38628922
%e A328656    8 |   135 | 0.37238098
%e A328656    9 |   159 | 0.35780768
%e A328656   10 |   186 | 0.32438582
%e A328656   11 |   212 | 0.29105188
%e A328656   12 |   315 | 0.24707528
%e A328656   13 |   363 | 0.24343744
%e A328656   14 |   453 | 0.23631515
%e A328656   15 |   693 | 0.18028720
%e A328656   16 |   922 | 0.13762601
%e A328656   17 |  1496 | 0.08925253
%e A328656   18 |  4765 | 0.04628960
%e A328656   19 |  6709 | 0.04209838
%e A328656   20 | 44555 | 0.04074628
%t A328656 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 A328656 If[kk <min, min = kk; AppendTo[aa, n]], {n, 1, 2000000}]; aa
%Y A328656 Cf. A114856, A254297, A255739, A255742, A326502, A329742.
%K A328656 nonn
%O A328656 1,2
%A A328656 _Artur Jasinski_, Jan 03 2020