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.
1, 2, 3, 5, 8, 10, 14, 20, 25, 28, 35, 64, 72, 92, 136, 160, 187, 213, 299, 316, 364, 454, 694, 923, 1497, 3778, 4766, 6710, 18860, 44556, 73998, 82553, 87762, 95249, 354770, 415588, 420892, 1115579, 8546951
Offset: 1
Keywords
Examples
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. 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. 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. 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. 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. 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.
Links
- Glen Pugh, The Riemann Hypothesis in a Nutshell.
Programs
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Mathematica
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
Formula
a(n) = A326502(n) + 1. - Artur Jasinski, Oct 24 2019
Extensions
a(38) from Arkadiusz Wesolowski, Nov 08 2015
a(39) from Artur Jasinski, Oct 24 2019
Comments