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

Author

Artur Jasinski, Jan 03 2020

Keywords

• nonn

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.

Examples

			   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
		

Links

• 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.

Crossrefs

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

Programs

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