A102522 -id:A102522 - cp's OEIS frontend

A102524 Continued fraction expansion of A102522.

0, 1, 1, 1, 21, 1, 3, 36, 3, 1, 1, 6, 2, 1, 1, 1, 1, 3, 1, 44, 1, 3, 3, 4, 13, 1, 4, 7, 1, 1, 3, 1, 3, 1, 4, 2, 1, 1, 1, 17, 5, 1, 9, 1, 1, 6, 1, 6, 12, 3, 5, 1, 1, 8, 1, 3, 30, 1, 4, 4, 1, 2, 2, 7, 1, 7, 1, 16, 7, 8, 7, 1, 51, 1, 1, 1, 240, 7, 1, 2, 1, 1, 1, 7, 4, 1, 10, 19, 3, 1, 6, 1, 22, 1, 6, 1, 1, 2, 5

Offset: 0

Gary W. Adamson and Robert G. Wilson v, Jan 13 2005

Increasing PQ's are: 1,21,36,44,51,240,298,999,2004,18156,130055, ...

Cf. A100060, A102522 (decimal expansion).

ContinuedFraction[ FromDigits[ {Join[{1}, Table[ If[ xx[[n]] > 0, 1, 0], {n, 1000}]], 0}, 2]]

Offset changed by Andrew Howroyd, Aug 04 2024

A100060 a(n)=1 if the n-th second difference of the imaginary parts of the nontrivial zeros of the Riemann zeta function is positive, otherwise a(n)=0.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1

Offset: 1

Gary W. Adamson, Oct 31 2004

nonn

Differences between zeta function gaps: increases are 1 and decreases are 0.

The ratios of the numbers of 0's to the number of 1's in the first 10^n differences are 0/1, 5/5, 50/50, 493/507, 4998/5002, 49949/50049, ...

			The first few positive t values of the zeros 1/2+i*t are (14.13..., 21.02..., 25.01..., 30.42..., 32.93..., 37.58..., 40.91..., 43.32...).
First differences are (6.88..., 3.98..., 5.41..., 2.51..., 4.65..., 3.33..., 2.40...).
Second differences are (-2.89..., 1.42..., -2.90..., 2.14..., -1.31..., -0.92...) which yields (0, 1, 0, 1, 0, 0, ...).

J. B. Conrey, A. Ghosh, D. Goldston, S. M. Gonek, and D. R. Heath-Brown, On the distribution of gaps between zeros of the Zeta-Function, Quart. J. Math. oxford 36 (1985) 43-51.
A. M. Odlyzko, On the distribution of spacings between zeros of the Zeta Function, Math. Comp. 48 (177) (1987) 273-308.
A. M. Odlyzko, Tables
Index entries for zeta function.

Cf. A102522, A102523.

zz = { (* the list of values in the link *) }; yy = Drop[zz, 1] - Drop[zz, -1]; Join[{1}, Table[ If[ yy[[n + 1]] > yy[[n]], 1, 0], {n, 104}]] (* Or *)
zz = { (* the list of values in the link *) }; yy = Drop[zz, 1] - Drop[zz, -1]; xx = Drop[yy, 1] - Drop[yy, -1]; Join[{1}, Table[ If[ xx[[n]] > 0, 1, 0], {n, 104}]] (* Robert G. Wilson v, Jan 14 2005 *)
Flatten[{1, (Sign[Differences[Differences[Im[ZetaZero[Range[106]]]]]] + 1)/2}] (* Mats Granvik, Jul 23 2015 *)

Corrected and extended by Robert G. Wilson v, Jan 13 2005

Edited by M. F. Hasler, Jul 27 2015

A123504 Sequence generated from the first nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 2

Gary W. Adamson, Oct 01 2006

nonn

A123505 records the lengths of runs. A123506 uses the second zero.

			a(8) = 1 since (1/8)^z = (0.353553..., angle 115.943... degrees).

Cf. A058303, A100060, A102522, A102523, A123505, A123506, A123507.

a[n_] := Boole[Arg[1/n^ZetaZero[1]] > 0]; Array[a, 100, 2] (* Amiram Eldar, May 31 2025 *)

t=1/2+solve(y=14,15,imag(zeta(1/2+y*I)))*I;
a(n)=arg(n^-t)>0 \\ Charles R Greathouse IV, Mar 10 2016

Extract argument from (1/n)^z, z = (1/2 + i*14.1347251417...). a(n) = 1 if the argument is between 0 and 180 degrees, and = 0 if otherwise (n = 2, 3, 4, ...).

More terms from Amiram Eldar, May 31 2025

A123505 Lengths of bit runs in A123504.

Original entry on oeis.org

1, 2, 1, 2, 2, 2, 3, 3, 5, 6, 7, 8, 11, 14, 17, 21, 26, 34, 41, 51, 65, 80, 101, 125, 157, 196, 245, 305, 381, 477, 595, 743, 927, 1159, 1448, 1807, 2258, 2819, 3521, 4397, 5492, 6859, 8565, 10698, 13361, 16685, 20839, 26026, 32503, 40593, 50697, 63315, 79074

Offset: 2

Gary W. Adamson, Oct 01 2006

nonn

A123507 records the bit runs of A123506 and uses the second zero in an analogous operation.

Record the numbers of consecutive bit runs of A123504, see example.

			A123504 = 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1...; therefore the numbers of bit runs are 1, 2, 1, 2, 2, 2, 3, ...

Cf. A100060, A102522, A102523, A123504, A123506, A123507.

Length /@ Split[Table[Boole[Arg[1/n^ZetaZero[1]] > 0], {n, 2, 10^6}]] (* Amiram Eldar, May 31 2025 *)

More terms from Amiram Eldar, May 31 2025

A123506 Sequence generated from the second nontrivial zero of the Riemann zeta function.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Gary W. Adamson, Oct 02 2006

nonn

A123504 performs an analogous set of operations using the first nontrivial zero. A123507 records the lengths of runs in this sequence.

Let z = (1/2 + i*t), t = 21.0220396387... (the second nontrivial Riemann zeta function zero). Perform (1/n)^z, (n = 2, 3, 4, ...) extracting the argument. If the argument is between 0 and 180 degrees, a(n) = 1, otherwise a(n) = 0.

			a(7) = 1 since (1/7)^z = (0.37796447..., angle 176.201... degrees) and the argument is between 0 and 180 degrees.

John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Plume - a Penguin Group, NY, 2003, pp. 198-199.

Cf. A065434, A102522, A102523, A123504, A123505, A123507.

a[n_] := Boole[Arg[1/n^ZetaZero[2]] > 0]; Array[a, 100, 2] (* Amiram Eldar, May 31 2025 *)

More terms from Amiram Eldar, May 31 2025

A123507 Lengths of bit runs in A123506.

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 2, 2, 3, 2, 4, 3, 5, 5, 5, 7, 8, 9, 11, 12, 14, 17, 19, 22, 26, 31, 34, 41, 47, 55, 64, 73, 86, 100, 115, 135, 156, 181, 210, 244, 283, 329, 383, 443, 516, 598, 695, 807, 936, 1088, 1263, 1467, 1703, 1978, 2297, 2666, 3097, 3595, 4176, 4848, 5630

Offset: 2

Gary W. Adamson, Oct 01 2006

nonn

The sequence uses operations based on the second nontrivial Riemann zero: (1/2 + i*t), t = 21.022039639... A123504 and A123505 use the first nontrivial zero.

Record the numbers of consecutive bit runs of A123506, see example.

			a(4) = 3 since A123506 = 0, 1, 1, 0, 1, 1, 1, ...

John Derbyshire, Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Plume - a Penguin Group, NY, 2003, pp. 198-199.

Cf. A100060, A102522, A102523, A123504, A123505, A123506.

Length /@ Split[Table[Boole[Arg[1/n^ZetaZero[2]] > 0], {n, 2, 10^6}]] (* Amiram Eldar, May 31 2025 *)

More terms from Amiram Eldar, May 31 2025

cp's OEIS Frontend

A102524 Continued fraction expansion of A102522.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A100060 a(n)=1 if the n-th second difference of the imaginary parts of the nontrivial zeros of the Riemann zeta function is positive, otherwise a(n)=0.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A123504 Sequence generated from the first nontrivial zero of the Riemann zeta function.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A123505 Lengths of bit runs in A123504.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A123506 Sequence generated from the second nontrivial zero of the Riemann zeta function.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Extensions

A123507 Lengths of bit runs in A123506.

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Programs

Mathematica

Extensions