cp's OEIS Frontend

A254297 Consider the nontrivial zeros of the Riemann zeta function on the critical line 1/2 + i*t and the gap, or first difference, between two consecutive such zeros; a(n) is the lesser of the two zeros at a place where the gap attains a new minimum.

%I A254297 #36 Mar 19 2025 15:06:15
%S A254297 1,2,3,5,8,10,14,20,25,28,35,64,72,92,136,160,187,213,299,316,364,454,
%T A254297 694,923,1497,3778,4766,6710,18860,44556,73998,82553,87762,95249,
%U A254297 354770,415588,420892,1115579,8546951
%N A254297 Consider the nontrivial zeros of the Riemann zeta function on the critical line 1/2 + i*t and the gap, or first difference, between two consecutive such zeros; a(n) is the lesser of the two zeros at a place where the gap attains a new minimum.
%C A254297 Since all zeros are assumed to be on the critical line, the gap, or first difference, between two consecutive zeros is measured as the difference between the two imaginary parts.
%C A254297 Inspired by A002410.
%C A254297 No other terms < 10000000. The minimum gap so far is 0.002323...
%H A254297 Glen Pugh, <a href="https://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html">The Riemann Hypothesis in a Nutshell</a>.
%F A254297 a(n) = A326502(n) + 1. - _Artur Jasinski_, Oct 24 2019
%e A254297 a(1)=1 since the first Riemann zeta zero, 1/2 + i*14.13472514... (A058303) has no previous zero, so its gap is measured from 0.
%e A254297 a(2)=2 since the second Riemann zeta zero, 1/2 + i*21.02203964... (A065434) has a gap of 6.887314497... which is less than the previous gap of ~14.13472514.
%e A254297 a(3)=3 since the third Riemann zeta zero, 1/2 + i*25.01085758... (A065452) has a gap of 3.988817941... which is less than ~6.887314497.
%e A254297 The fourth Riemann zeta zero, 1/2 + i*30.42487613... (A065453) has a gap of 5.414018546... which is not less than ~6.887314497 and therefore is not in the sequence.
%e A254297 a(4)=5 since the fifth Riemann zeta zero, 1/2 + i*32.93506159... (A192492) has a gap of 2.510185462... which is less than ~3.988817941.
%e A254297 a(5)=8 since the eighth Riemann zeta zero, 1/2 + i*43.32707328...  has a gap of 2.408354269... which is less than ~2.510185462.
%t A254297 k = 1; mn = Infinity; y = 0; lst = {}; While[k < 10001, z = N[ Im@ ZetaZero@ k, 64]; If[z - y < mn, mn = z - y; AppendTo[lst, k]]; y = z; k++]; lst
%Y A254297 Cf. A002410, A100060, A161914, A117538, A153595, A326502.
%K A254297 nonn
%O A254297 1,2
%A A254297 _Robert G. Wilson v_, Jan 27 2015
%E A254297 a(38) from _Arkadiusz Wesolowski_, Nov 08 2015
%E A254297 a(39) from _Artur Jasinski_, Oct 24 2019