

A208436


Indices of maximal gaps between consecutive nontrivial zeros of the Riemann zeta function.


4



1, 3, 8, 14, 33, 64, 79, 126, 183, 379, 795, 1935, 2292, 3296, 4805, 6620, 15323, 19187, 20105, 36719, 46589, 185013, 220571, 259501, 516200, 880694, 1493008, 1663325, 1793281, 3206674, 6488753, 14145077, 22653912, 33742399, 65336924, 70354407, 81805537, 110280572, 129842508, 298466597, 566415148
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OFFSET

1,2


COMMENTS

Values are conjectural: in principle exact values can be computed, but the bound involves a triple logarithm (or double logarithm under RH) and so is computationally infeasible.
Corresponding gap sizes are (to 7 decimal places) 6.887314, 5.414019, 4.678078, 4.280766, 3.860924, 3.499560, 3.249442, 3.235864, 3.011009, 2.980888, 2.594919, 2.589789, 2.539380, 2.406590, 2.279428, 2.194404, 2.176083, 2.098284, 2.064198, 2.042407, 2.024333, 1.966653, 1.844023, 1.804885, 1.798398, 1.779155, 1.754010, 1.696635, 1.688765, 1.686034, 1.580157, 1.567382, 1.525555, 1.521410, 1.488847, 1.479976, 1.432771, 1.422617, 1.420599, 1.413245, 1.393242, ....
Goldston & Gonek show, on the Riemann Hypothesis, that the gap between the zero 1/2 + ix and the following zero is at most Pi(1 + o(1))/log log x. For illustrative purposes, if o(1) is taken to be zero, it would suffice to check up to height 345.9 to verify a(24) with gap size 1.804... but over 10^10 for gap size 1.  Charles R Greathouse IV, Oct 22 2012


REFERENCES

E. C. Titchmarsh, The theory of the Riemann zetafunction, 1986.


LINKS

Table of n, a(n) for n=1..41.
D. A. Goldston and S. M. Gonek, A note on S(t) and the zeros of the Riemann zetafunction, Bull. London Math. Soc. 39 (2007), pp. 482486.
LMFDB, Zeros of ΞΆ(s)
Andrew Odlyzko, Tables of zeros of the Riemann zeta function
Index entries for zeta function


EXAMPLE

The first four nontrivial zeros of the zeta function are at
0.5 + 14.13472...i, 0.5 + 21.02203...i, 0.5 + 25.01085...i, and 0.5 + 30.42487...i with gaps 6.88731..., 3.98882..., and 5.41402.... No gap is larger than the first, so a(1) = 1. The third gap is larger than all gaps later than the first gap, so a(2) = 3.


CROSSREFS

Cf. A161914, A002410.
Sequence in context: A298612 A168155 A005735 * A297015 A135872 A242245
Adjacent sequences: A208433 A208434 A208435 * A208437 A208438 A208439


KEYWORD

nonn,hard,nice


AUTHOR

Charles R Greathouse IV, Feb 26 2012


EXTENSIONS

a(27)a(41) from Andrey V. Kulsha, Aug 27 2012


STATUS

approved



