

A088750


a(n) is the index of that zero of the Riemann zetafunction on the same line as the Gram point g(n2). It is only welldefined if the Riemann hypothesis is true.


4



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
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OFFSET

1,2


COMMENTS

The zeros of the Riemann zetafunction are numbered. The ordinates being 0<gamma_1 < gamma_2<gamma_3 < ... The sequence refers to the number of the zero.
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 na(n)<C log n. By a theorem of Speiser the sequence is welldefined if and only if the hypothesis of Riemann is true. Some relations with the sequence 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.


LINKS



EXAMPLE

a(9)=10 because the Gram point g(7)=g(92) is on the same sheet Im zeta(s)=0 that the tenth nontrivial zero of Riemann zeta function.


CROSSREFS



KEYWORD

hard,nonn


AUTHOR



STATUS

approved



