A088749 -id:A088749 - cp's OEIS frontend

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

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

A088750 a(n) is the index of that zero of the Riemann zeta function on the same line as the Gram point g(n-2). It is only well-defined if the Riemann hypothesis is true.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 9, 11, 13, 12, 14, 16, 15, 17, 18, 20, 19, 21, 24, 22, 23, 25, 27, 26, 28, 29, 32, 30, 31, 33, 35, 34, 36, 37, 40, 38, 39, 41, 44, 42, 43, 45, 46, 48, 47, 49, 50, 53, 51, 52, 54, 55, 57, 56, 58

Offset: 1

Juan Arias-de-Reyna, Oct 15 2003

The zeros of the Riemann zeta function are numbered. The ordinates being 0

To make the relation between zeros and Gram points bijective we must associate the Gram points on a parallel line with the zero on the next parallel line above it. n->a(n) is a bijection of the natural numbers. For some absolute constant C and every n we have |n-a(n)|A088749 that appear to be true for the first terms are not true in general. The sequence is given with some mistakes in the reference arXiv:math.NT/0309433.

The only way I know to obtain the sequence is to draw the curves Re zeta(s)=0 and Im zeta(s)=0.

			a(9)=10 because the Gram point g(7)=g(9-2) is on the same sheet Im zeta(s)=0 that the tenth nontrivial zero of Riemann zeta function.

Juan Arias-de-Reyna, Table of n, a(n) for n = 1..79
J. Arias-de-Reyna, X-Ray of Riemann zeta-function, arXiv:math/0309433 [math.NT], 2003.
Juan Arias-de-Reyna, Further notes on A088750, September 2018.
A. Speiser, Geometrisches zur Riemannschen Zetafunktion, Math. Ann., Vol. 110 (1934), pp. 514-521.

Cf. A088749.

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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.

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A088750 a(n) is the index of that zero of the Riemann zeta function on the same line as the Gram point g(n-2). It is only well-defined if the Riemann hypothesis is true.

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cp's OEIS Frontend

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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References

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Crossrefs

Programs

PARI

Extensions

A088750 a(n) is the index of that zero of the Riemann zeta function on the same line as the Gram point g(n-2). It is only well-defined if the Riemann hypothesis is true.

Views

Author

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Examples

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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.