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.

1, 2, 4, 9, 13, 27, 34, 135, 159, 186, 212, 315, 363, 453, 693, 922, 1496, 4765, 6709, 44555, 73997, 82552, 87761, 95248, 415587, 420891, 1115578, 8546950, 24360732, 41820581, 1048449114, 3570918901, 35016977796

Offset

1,2

Comments

No more records up to k = 103800788359.

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

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

Conjectural next terms: 1217992279429, 4088664936219.

Links

Table of n, a(n) for n=1..33.

Xavier Gourdon, The 1013 first zeros of the Riemann Zeta function, and zeros computation at very large height, 2004.

David Platt, Results computation of the smallest relative gaps between successive zeta zeros, 2020.

Example

   n |  a(n) |    d(n)
  ---+-------+------------
   1 |     1 | 0.88871193
   2 |     2 | 0.76669277
   3 |     4 | 0.63017799
   4 |     9 | 0.57239954
   5 |    13 | 0.53062398
   6 |    27 | 0.52634271
   7 |    34 | 0.38628922
   8 |   135 | 0.37238098
   9 |   159 | 0.35780768
  10 |   186 | 0.32438582
  11 |   212 | 0.29105188
  12 |   315 | 0.24707528
  13 |   363 | 0.24343744
  14 |   453 | 0.23631515
  15 |   693 | 0.18028720
  16 |   922 | 0.13762601
  17 |  1496 | 0.08925253
  18 |  4765 | 0.04628960
  19 |  6709 | 0.04209838
  20 | 44555 | 0.04074628

Mathematica

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

Crossrefs

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

Sequence in context: A114885 A049793 A090942 * A085901 A077224 A059447

Adjacent sequences: A328653 A328654 A328655 * A328657 A328658 A328659

Keyword

nonn

Author

Artur Jasinski, Jan 03 2020

Status

approved