A117538 -id:A117538 - cp's OEIS frontend

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

Robert G. Wilson v, Jan 27 2015

nonn

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...

			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.

Glen Pugh, The Riemann Hypothesis in a Nutshell.

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

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

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

a(38) from Arkadiusz Wesolowski, Nov 08 2015

a(39) from Artur Jasinski, Oct 24 2019

A117536 Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i2(Pi/log(2))*t)) for increasing real t.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270, 342, 422, 441, 494, 742, 764, 935, 954, 1012, 1106, 1178, 1236, 1395, 1448, 1578, 2460, 2684, 3395, 5585, 6079, 7033, 8269, 8539, 11664, 14348, 16808, 28742, 34691

Offset: 0

Gene Ward Smith, Mar 27 2006

These correspond to increasing peaks of the absolute value of the Riemann zeta function along the critical line. If Z'(s)=0 is a positive zero of the derivative of Z, then |Z(s)| is the peak value.
The fractional parts of these values are not randomly distributed; r = log(2) * s(n) / (2*Pi) shows a very strong tendency to be near an integer.
It would be interesting to have theorems on the distribution of the fractional part of the "r" above, for which the Riemann hypothesis would surely be needed. It would be particularly interesting to know if the absolute value's fractional part is constrained to be less than some bound, such as 0.25. This computation could be pushed much farther by someone using a better algorithm, for instance the Riemann-Siegel formula and better computing resources. The computations were done using Maple's accurate but very slow zeta function evaluation. They are correct as far as they go, but do not go very far. The terms of the sequence have an interpretation in terms of music theory; the terms which appear in it, 12, 19, 22 and so forth, are equal divisions of the octave which do relatively well approximating intervals given by rational numbers with small numerators and denominators.
This sequence was extended by examining the peaks of |zeta(0.5+xi)| between each the first million zeros of the zeta function. These record peaks occur between zeros that are relatively far apart. The fractional part of r decreases as the magnitude of r increases. - T. D. Noe, Apr 19 2010

			The function f(m) = |zeta(1/2 + i*2*(Pi/log(2))*m)| has a local maximum f(m') ~ 3.66 at m' ~ 5.0345, which corresponds to a(5)=round(m)=5. The peak at f(6.035) ~ 2.9 is smaller, and after two more smaller local maxima, there is a larger peak at f(6.9567) ~ 4.167, whence a(6)=7.

H. M. Edwards, Riemann's Zeta-Function, Academic Press, 1974.
K. Ramachandra, On the Mean-Value and Omega-Theorems for the Riemann Zeta-Function, Springer-Verlag, 1995.
E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986.

Andrew Odlyzko, Tables of the zeros of the Riemann zeta function.
Wikipedia, Z function
Index entries for zeta function.

Cf. A117537, A117538, A117539, A079630, A088749, A088750, A054540.

{my(c=I/log(2)*2*Pi,f(n)=abs(zeta(.5+n*c)), m=0,
find(x,d,e=1e-6)=my(y=f(x)); while(y<(y=f(x+=d)) || eM. F. Hasler, Jan 26 2012

Extended by T. D. Noe, Apr 19 2010

A117559 Equal divisions of the octave of decreasing fifteen-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 7, 8, 24, 58, 111, 130, 224, 270, 494, 2190, 2684, 5585, 6079, 14618, 20203, 81860, 96478

Offset: 0

Gene Ward Smith, Mar 28 2006

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest.

Wikipedia, Tonality diamond

Cf. A117554, A117555, A117556, A117557, A117559, A005664, A117536, A117538.

A117555 Equal divisions of the octave of decreasing seven-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 3, 4, 12, 22, 27, 31, 99, 171, 3125, 6691, 11664, 18355, 84814, 103169

Offset: 0

Gene Ward Smith, Mar 28 2006

We may define the n-limit Pepper ambiguity, for any odd number greater than one n, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest. In the 7-limit, that means we look at the ratios of the errors for the nearest approximations to 3/2, 5/4, 5/3, 7/4, 7/5 and 7/6 to the next nearest.

Wikipedia, Tonality diamond

Cf. A117554, A117556, A117557, A117558, A117559, A005664, A117536, A117538.

A117556 Equal divisions of the octave of decreasing nine-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 4, 5, 12, 19, 31, 41, 99, 171, 3125, 11664, 18355, 84814, 103169

Offset: 0

Gene Ward Smith, Mar 28 2006

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest. In the 9-limit, that means we look at the ratios of the errors for the nearest approximations to 3/2, 5/4, 5/3, 7/4, 7/5, 7/6, 9/8, 9/7 and 9/5 to the next nearest.

Wikipedia, Tonality diamond

Cf. A117554, A117555, A117557, A117558, A117559, A005664, A117536, A117538.

A117537 Locations of the midpoints of consecutive zeros of the Riemann zeta function on the critical line with increasingly large normalized spacing.

Original entry on oeis.org

2, 3, 5, 7, 12, 19, 31, 46, 53, 72, 270, 311, 954, 1178, 1308, 1395, 1578, 3395, 4190

Offset: 0

Gene Ward Smith, Mar 27 2006

Equivalently, locations of consecutive real zeros of the Z function. If t and s are consecutive zeros of the Z function, we define their normalized spacing as (s-t)*log((s+t)/(4*Pi)). The sequence above is found by taking r = log(2)(s+t)/(4*Pi) and rounding to the nearest integer. These values r have a marked tendency to be close to integer values and all of the terms of the above sequence are actually contained in the intervals [s, t]*log(2)/(2*Pi).

So far as the first 100000 zeros take us, the integers of the above sequence actually fall inside the normalized intervals of zeros of Z; that is, they fall between two zeros of Z(2*Pi*t/log(2)). It would be a worthwhile project to push this computation far enough to find a counterexample. The integers above, while slightly less clearly linked to music than A117536 and A117538, are nevertheless very clearly closely related to equal divisions of the octave. Large gaps between consecutive zeros, in other words, seem to correspond to good scale divisions, though less exactly than peak values or high integrals do.

Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
A. Ivic (1985). The Riemann Zeta Function, John Wiley & Sons. ISBN 0-471-80634-X.
Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986

A. M. Odlyzko, On the distribution of spacings between zeros of the zeta function, Math. Comp., 48 (1987), 273-308.
A. M. Odlyzko, The first 100,000 zeros of the Riemann zeta function, accurate to within 3*10^(-9)
Wikipedia, Z Function

Cf. A117536, A117538, A117539.

A117539 Integrals of the absolute value of the Z function between successive zeros greater than or equal to the integral corresponding to 12. If we define the normalized Z function by z(x) = Z(2Pix/log(2)), then the 33rd and 34th zeros are approximately 11.82 and 12.25. Integrating |z(x)| between these values gives a quantity I and the above sequence is defined as the midpoints of all successive zeros of z(x) such that the integral of |z(x)| is greater than or equal to I.

Original entry on oeis.org

12, 19, 31, 41, 46, 53, 58, 65, 72, 77, 87, 94, 99, 103, 111

Offset: 0

Gene Ward Smith, Mar 27 2006

The reason for the choice of 12 as a starting point is from musical practice; 12 is the standard equal division of the octave of Western music. The subsequent values where this integral is greater than it is for 12 are also equal divisions. While all the values tabulated are such that the integer of the integer sequence is actually contained in the interval between two successive zeros, it must eventually happen that a counterexample would be found. Another interesting question is the density of this sequence; it is not clear if it is increasing in density or not.

Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986

Andrew Odlyzko, The first 100,000 zeros of the Riemann zeta function, accurate to within 3*10^(-9)
Wikipedia, Z function

Cf. A117536, A117537, A117538, A054540.

A117558 Equal divisions of the octave of decreasing thirteen-limit Pepper ambiguity.

Original entry on oeis.org

1, 2, 7, 8, 24, 37, 46, 58, 130, 198, 224, 270, 494, 1506, 2684, 5585, 6079, 14618, 20203, 81860, 87939, 96478

Offset: 0

Gene Ward Smith, Mar 28 2006

We may define the n-limit Pepper ambiguity, for any odd number n greater than one, as the maximum of the ratios of the errors of the nearest approximation to the members of the n-limit tonality diamond to the next nearest.

Wikipedia, Tonality diamond

Cf. A117554, A117555, A117556, A117557, A117559, A005664, A117536, A117538.

cp's OEIS Frontend

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.

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A117536 Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i*2*(Pi/log(2))*t)) for increasing real t.

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A117559 Equal divisions of the octave of decreasing fifteen-limit Pepper ambiguity.

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A117555 Equal divisions of the octave of decreasing seven-limit Pepper ambiguity.

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A117556 Equal divisions of the octave of decreasing nine-limit Pepper ambiguity.

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A117537 Locations of the midpoints of consecutive zeros of the Riemann zeta function on the critical line with increasingly large normalized spacing.

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A117558 Equal divisions of the octave of decreasing thirteen-limit Pepper ambiguity.

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A117536 Nearest integer to locations of increasingly large peaks of abs(zeta(0.5 + i2(Pi/log(2))*t)) for increasing real t.