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.

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

Author

Juan Arias-de-Reyna, Oct 15 2003

Keywords

• hard
• nonn

Comments

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.

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A088749.

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