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A329742 Indices n of Riemann zeta zeros for successive records of the normalized delta defined as d(n) = (z(n+1)-z(n))*(log(z(n)/(2Pi))/(2Pi)) where z(n) is the imaginary part of the n-th Riemann zero.

%I A329742 #42 Mar 30 2020 10:31:06
%S A329742 1,3,5,8,14,25,33,64,126,213,256,379,1704,1935,2292,8571,10942,12347,
%T A329742 13298,15323,36719,46589,103715,185013,880694,1493008,3206674,
%U A329742 12534781,14145077,22653912,24246374,33742399,65336924,298466597,566415148,1938289664,2122614029,4020755339,4219726754,16265396008,17003807756
%N A329742 Indices n of Riemann zeta zeros for successive records of the normalized delta defined as d(n) = (z(n+1)-z(n))*(log(z(n)/(2Pi))/(2Pi)) where z(n) is the imaginary part of the n-th Riemann zero.
%C A329742 No more records up to n = 103800788359.
%C A329742 d(17003807756) = 4.3018209763411.
%C A329742 Successive records occur when gaps between two successive zeros are large.
%C A329742 Recent record of normalized delta computed by Hiary at 2011 occurs for n=436677148707320393224019748290912 where d(n) = 5.77979.
%C A329742 Conjectural next term: 77528045597.
%C A329742 Indices of zeros for successive minimal records of the normalized delta see A328656.
%H A329742 Artur Jasinski, <a href="/A329742/b329742.txt">Table of n, a(n) for n = 1..41</a>
%H A329742 Ghaith Ayesh Hiary, <a href="http://dx.doi.org/10.4007/annals.2011.174.2.4">Fast methods to compute the Riemann zeta function</a>, Ann. of Math. (2) 174 (2011), no. 2, 891-946. MR 2831110 (2012g:11154).
%H A329742 David Platt, <a href="/A329742/a329742.txt">Results computation of the largest relative gaps between successive zeta zeros</a>, 2020.
%e A329742    n |   a(n)  |  d(n)
%e A329742   ---+---------+---------
%e A329742    1 |       1 | 0.88871
%e A329742    2 |       3 | 1.19034
%e A329742    3 |       5 | 1.22634
%e A329742    4 |       8 | 1.43763
%e A329742    5 |      14 | 1.54672
%e A329742    6 |      25 | 1.55244
%e A329742    7 |      33 | 1.74300
%e A329742    8 |      64 | 1.83656
%e A329742    9 |     126 | 1.95400
%e A329742   10 |     213 | 1.95626
%e A329742   11 |     256 | 1.99205
%e A329742   12 |     379 | 2.20138
%e A329742   13 |    1704 | 2.20198
%e A329742   14 |    1935 | 2.45843
%e A329742   15 |    2292 | 2.46772
%e A329742   16 |    8571 | 2.48347
%e A329742   17 |   10942 | 2.50594
%e A329742   18 |   12347 | 2.50648
%e A329742   19 |   13298 | 2.52517
%e A329742   20 |   15323 | 2.67728
%e A329742   21 |   36719 | 2.76188
%e A329742   22 |   46589 | 2.80523
%e A329742   23 |  103715 | 2.83121
%e A329742   24 |  185013 | 3.11058
%e A329742   25 |  880694 | 3.21426
%e A329742   26 | 1493008 | 3.30347
%t A329742 prec = 30; max = 0; aa = {}; Do[kk = N[Im[(ZetaZero[n + 1] - ZetaZero[n])],prec] (Log[N[Im[ZetaZero[n]], prec]/(2 Pi)]/(2 Pi));
%t A329742 If[kk > max, max = kk; AppendTo[aa, n]], {n, 1, 2000000}]; aa
%Y A329742 Cf. A114856, A254297, A255739, A255742, A326502.
%K A329742 nonn
%O A329742 1,2
%A A329742 _Artur Jasinski_, Nov 20 2019
%E A329742 a(27)-a(41) computed by David Platt, Jan 03 2020