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

1,2

Comments

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.

Inspired by A002410.

No other terms < 10000000. The minimum gap so far is 0.002323...

Links

Table of n, a(n) for n=1..39.

Glen Pugh, The Riemann Hypothesis in a Nutshell.

Formula

a(n) = A326502(n) + 1. - Artur Jasinski, Oct 24 2019

Example

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.

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

Crossrefs

Cf. A002410, A100060, A161914, A117538, A153595, A326502.

Sequence in context: A368052 A022955 A363076 * A306972 A087279 A246346

Adjacent sequences: A254294 A254295 A254296 * A254298 A254299 A254300

Keyword

nonn

Author

Robert G. Wilson v, Jan 27 2015

Extensions

a(38) from Arkadiusz Wesolowski, Nov 08 2015

a(39) from Artur Jasinski, Oct 24 2019

Status

approved