**RIEMANN'S HYPOTHESIS AND THE SHAPE OF THE EARTH II**Highest zeta zero ever computed:

t ≈ 81029194732694548890047854481676712.9879 ( n = 10

^{36} + 4242063737401796198).

https://arxiv.org/pdf/1607.00709.pdf1273315917355388788579148020712834 x 63 = 80218902793389493680486325304908542

1273315917355388788579148020712834 x 0.63636363 = 810291939305055209561529176768134.23182742

**81,029,194,732,694,548,890,047,854,481,676,676.23182742**

81,029,194,732,694,548,890,047,854,481,676,739.8681910581,029,194,732,694,548,890,047,854,481,676,692.40916072

(+16.1773333)

81,029,194,732,694,548,890,047,854,481,676,704.47514472

(+12.065984)

81,029,194,732,694,548,890,047,854,481,676,713.47347282

(+8.9983281)

81,029,194,732,694,548,890,047,854,481,676,709.78410162

(+5.3089569)

81,029,194,732,694,548,890,047,854,481,676,710.72208732

(+0.9379857)

81,029,194,732,694,548,890,047,854,481,676,711.42159955

(+0.69951223)

81,029,194,732,694,548,890,047,854,481,676,711.943267789

(+0.521668239)

81,029,194,732,694,548,890,047,854,481,676,712.332307089

(+0.3890393)

81,029,194,732,694,548,890,047,854,481,676,712.622437039

(+0.29012995)

81,029,194,732,694,548,890,047,854,481,676,712.838804339

(+0.2163673)

81,029,194,732,694,548,890,047,854,481,676,723.69085805

(-16.177333)

81,029,194,732,694,548,890,047,854,481,676,711.62487405

(-12.065984)

81,029,194,732,694,548,890,047,854,481,676,716.5720035

(-7.11885455)

81,029,194,732,694,548,890,047,854,481,676,715.3142453

(-1.2577582)

81,029,194,732,694,548,890,047,854,481,676,714.37625955

(-0.93798575)

81,029,194,732,694,548,890,047,854,481,676,713.6767473

(-0.69951225)

81,029,194,732,694,548,890,047,854,481,676,713.15507905

(-0.52166825)

**Very interesting comments on the S(t) function:**https://arxiv.org/pdf/1407.4358.pdf (

**page 46**)

To test any hypotheses regarding S(t), the five element subdivision algorithms should be used around the zeta zero t = 10

^{10,000}, since the 63.636363636363... interval has to be shifted/translated only using arbitrary-precision arithmetic ([10

^{10,000}/63.6363636] x 63.63636363, where [ x ] denotes the integral part of x). To detect the correct number of zeros in the interval [10

^{10,000}/63.63636363...] x 63.63636363, [10

^{10,000}/63.63636363...] x 63.63636363 + 63.636363636363, Gram points, França-LeClair points, Backlund's method should be utilized, and then simply use the five element subdivision algorithms to compute the zeta zeros within that interval, and discover how S(t) behaves at that height (for 10

^{10,000}, the average spacing is 0.0002729).

There is a way to prove the Riemann hypothesis: to check if any of the zeta zeros can be obtained using the four points (or their subsequent subdivisions) derived from the five element subdivision algorithm for a previous zero.

That is, if z

_{1} is a zeta zero, then z

_{2} = z

_{1} + n⋅63.63636363... (for z

_{1} a certain four subdivision points were used, and for z

_{2}, further subdivisions/partitions from the first value (z

_{1}) are used to attain the final value; at the height t = z

_{2}, there is a much greater density of zeros that at the height t = z

_{1}).

Notwithstanding the increasing density of the zeros, it would be interesting to check this hypothesis at least up to the height t = 10

^{13}.

A brief example: for z = 14.134725... (the very first zeta zero).

z

_{2} = 14.134725... + n⋅63.63636363... = x

n = 45

x = 2877.771

http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1z = 2877.5466 (the interval around this value contains a large gap)

n = 450

x = 28650.498

z = 28650.784 (contains a Lehmer pair)

n = 286

x = 18214.1345

z = 18214.358 (contains a Lehmer pair)

n = 535

x = 33995.176

z = 33996.251 (contains a Lehmer pair)

n = 1018

x = 64795.888

z = 64795.6137 (contains a Lehmer pair)

z

_{2} = 21.02204 + n⋅63.63636363... = x

n = 45

x = 2884.658

z = 2884.347 (contains a Lehmer pair)

n = 450

x = 28657.3854

z = 28657.477 (contains a Lehmer pair)

n = 286

x = 18221.023

z = 18221.043 (a close match, also contains a large gap)

n = 535

x = 34002.84

z = 34003.13 (contains a Lehmer pair)

n = 1018

x = 64802.84

z = 64802.88 (a close match)

https://arxiv.org/pdf/1502.06003.pdfFormula used to calculate the zeta zeros:

ϑ(t

_{n}) + lim

_{δ→0+} arg ζ(1/2 + δ + it

_{n}) = (n - 3/2)π

t

_{1} = z

_{1}t

_{2} = z

_{1} + k⋅63.63636363...

ϑ(z

_{1}) + lim

_{δ→0+} arg ζ(1/2 + δ + iz

_{1}) = (n - 3/2)π

ϑ(z

_{1} + k⋅63.63636363...) + lim

_{δ→0+} arg ζ(1/2 + δ + i{z

_{1} + k⋅63.63636363...}) = (l - 3/2)π

Even if a relationship could be found between k, l and n, we would have two simultaneous nonlinear equations featuring three unknowns (z

_{1}, k and n or l).