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Search: id:A117536
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| A117536 |
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These are the locations of the increasingly larger peaks of the absolute value of the Riemann zeta function along the critical line. Equivalently, the locations of the increasingly large peaks of the absolute value of the Z function for increasing real t. If Z'(s)=0 is a positive zero of the derivative of Z, then |Z(s)| is the peak value. We renormalize s by r = ln(2) s /2 pi, and round to the nearest integer to get the terms of the sequence. The fractional parts of these values are not randomly distributed; r shows a very strong tendency to be near an integer. |
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+0
8
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| 0, 1, 2, 3, 4, 5, 7, 10, 12, 19, 22, 27, 31, 41, 53, 72, 99, 118, 130, 152, 171, 217, 224, 270
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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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 fractional part was 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.
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REFERENCES
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Edwards, H. M., Riemann's Zeta-Function, Academic Press, 1974
Ramachandra, K., On the Mean-Value and Omega-Theorems for the Riemann Zeta-Function, Springer-Verlag, 1995
Titchmarsh, E. C., The Theory of the Riemann Zeta-Function, second revised (Heath-Brown) edition, Oxford University Press, 1986
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LINKS
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Odlyzko's tables of the zeros of the Riemann zeta function
Wikipedia article on the Z function
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CROSSREFS
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Cf. A117537, A117538, A117539, A079630, A088749, A088750, A054540.
Sequence in context: A132975 A145977 A050729 this_sequence A104665 A094018 A071682
Adjacent sequences: A117533 A117534 A117535 this_sequence A117537 A117538 A117539
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KEYWORD
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hard,nonn
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AUTHOR
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Gene Ward Smith (genewardsmith(AT)gmail.com), Mar 27 2006
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