A208436 Indices of maximal gaps between consecutive nontrivial zeros of the Riemann zeta function.

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

Offset: 1

Author

Charles R Greathouse IV, Feb 26 2012

Keywords

• nonn
• hard
• nice

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

Littlewood proves an unconditional version: there is some constant Y such that the gap between the zero x + i*y and the following zero is at most 32/log log log y for all y > Y. Hall & Hayman improve the above constant from 32 to Pi/2 + o(1). Even with this latter improvement, verifying a gap of size 1 with this formula (even dropping the o(1)) would take a height above 10^53. - Charles R Greathouse IV, Jun 04 2021

Ivić improves the Goldston & Gonek constant to Pi/2. - Charles R Greathouse IV, Jun 23 2021

Bui & Milinovich, improving on Bredberg, prove that, for large enough T, there is a gap of length 6.36*Pi/log T between T and 2T (that is, 3.18 times the length of the average gap). This seems rather sharp around values where we're computing this sequence. - Charles R Greathouse IV, Jun 24 2021

Simonič (2018) shows that a(1) = 1, see the proof of Lemma 3 around (8). - Charles R Greathouse IV, Jul 07 2021

Simonič (2022) shows that, under the Riemann hypothesis, it suffices to check up to height 10^2465 to prove that the values of a(2)-a(40) are as stated. a(41) requires checking to 10^2588. Because these values are so high it is not currently feasible to prove more terms of the sequence, even under RH. - Charles R Greathouse IV, Mar 28 2022

Examples

			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.
		

References

• R. R. Hall and W. K. Hayman, Hyperbolic distance and distinct zeros of the Riemann zeta-function in small regions, Journal für die reine und angewandte Mathematik vol. 526 (2000), pp. 35-59.
• J. E. Littlewood, Two notes on the Riemann zeta-function, Proceedings of the Cambridge Philosophical Society Vol. 22, No. 3 (1924), pp. 234-242.
• E. C. Titchmarsh, The theory of the Riemann zeta-function, 1986.

Links

• H. M. Bui and M. B. Milinovich, Gaps between zeros of the Riemann zeta-function, 2017 preprint. arXiv:1410.3635 [math.NT]
• Johan Bredberg, Large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function, 2011 preprint. arXiv:1101.3197 [math.NT]
• D. A. Goldston and S. M. Gonek, A note on S(t) and the zeros of the Riemann zeta-function, Bull. London Math. Soc. 39 (2007), pp. 482-486.
• Aleksandar Ivić, Some remarks on the differences between ordinates of consecutive zeta zeros, Funct. Approx. Comment. Math. 58:1 (2018), pp. 23-35. arXiv:1610.01317 [math.NT]
• LMFDB, Zeros of ζ(s)
• Andrew Odlyzko, Tables of zeros of the Riemann zeta function
• Aleksander Simonič, Lehmer pairs and derivatives of Hardy's Z-function, Journal of Number Theory, Vol. 184 (March 2018), pp. 451-460. arXiv:1612.08627 [math.NT]
• Aleksander Simonič, On explicit estimates for S(t), S1(t), and ζ(1/2 + it) under the Riemann Hypothesis, Journal of Number Theory, Vol. 231 (February 2022), pp. 464-491. arXiv:2010.13307 [math.NT]
• Index entries for zeta function

Crossrefs

Cf. A161914, A002410.

Extensions

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