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Search: id:A088750
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| A088750 |
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a(n) = number of the 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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+0
3
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| 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
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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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)|<C log n. By a theorem of Speiser the sequence is well-defined if and only if the hypothesis of Riemann is true. Some relations with the sequence A088749 that appear to be true for the first temrs are not true in general. The sequence is given with some mistakes in the reference arXiv:math.NT/0309433.
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REFERENCES
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A. Speiser, Geometrisches zur Riemannschen Zetafunktion, Math. Ann., Vol. 110 (1934), pp. 514-521
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LINKS
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J. Arias-de-Reyna, X-Ray of Riemann zeta-function
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EXAMPLE
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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.
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MAPLE
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The only way I know to obtain the sequence is to draw the curves Re zeta(s)=0 and Im zeta(s)=0.
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CROSSREFS
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Cf. A088749.
Sequence in context: A095903 A166277 A145342 this_sequence A056018 A087465 A056017
Adjacent sequences: A088747 A088748 A088749 this_sequence A088751 A088752 A088753
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KEYWORD
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hard,nonn
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AUTHOR
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J. Arias-de-Reyna (arias(AT)us.es), Oct 15 2003
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