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.

1, 3, 5, 8, 14, 25, 33, 64, 126, 213, 256, 379, 1704, 1935, 2292, 8571, 10942, 12347, 13298, 15323, 36719, 46589, 103715, 185013, 880694, 1493008, 3206674, 12534781, 14145077, 22653912, 24246374, 33742399, 65336924, 298466597, 566415148, 1938289664, 2122614029, 4020755339, 4219726754, 16265396008, 17003807756

Offset: 1

Author

Artur Jasinski, Nov 20 2019

Keywords

• nonn

Comments

No more records up to n = 103800788359.

d(17003807756) = 4.3018209763411.

Successive records occur when gaps between two successive zeros are large.

Recent record of normalized delta computed by Hiary at 2011 occurs for n=436677148707320393224019748290912 where d(n) = 5.77979.

Conjectural next term: 77528045597.

Indices of zeros for successive minimal records of the normalized delta see A328656.

Examples

			   n |   a(n)  |  d(n)
  ---+---------+---------
   1 |       1 | 0.88871
   2 |       3 | 1.19034
   3 |       5 | 1.22634
   4 |       8 | 1.43763
   5 |      14 | 1.54672
   6 |      25 | 1.55244
   7 |      33 | 1.74300
   8 |      64 | 1.83656
   9 |     126 | 1.95400
  10 |     213 | 1.95626
  11 |     256 | 1.99205
  12 |     379 | 2.20138
  13 |    1704 | 2.20198
  14 |    1935 | 2.45843
  15 |    2292 | 2.46772
  16 |    8571 | 2.48347
  17 |   10942 | 2.50594
  18 |   12347 | 2.50648
  19 |   13298 | 2.52517
  20 |   15323 | 2.67728
  21 |   36719 | 2.76188
  22 |   46589 | 2.80523
  23 |  103715 | 2.83121
  24 |  185013 | 3.11058
  25 |  880694 | 3.21426
  26 | 1493008 | 3.30347
		

Links

• Artur Jasinski, Table of n, a(n) for n = 1..41
• Ghaith Ayesh Hiary, Fast methods to compute the Riemann zeta function, Ann. of Math. (2) 174 (2011), no. 2, 891-946. MR 2831110 (2012g:11154).
• David Platt, Results computation of the largest relative gaps between successive zeta zeros, 2020.

Crossrefs

Cf. A114856, A254297, A255739, A255742, A326502.

Programs

Mathematica

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));
If[kk > max, max = kk; AppendTo[aa, n]], {n, 1, 2000000}]; aa

Extensions

a(27)-a(41) computed by David Platt, Jan 03 2020