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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.

%I A088750 #28 Mar 27 2024 20:11:00
%S A088750 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,
%T A088750 26,28,29,32,30,31,33,35,34,36,37,40,38,39,41,44,42,43,45,46,48,47,49,
%U A088750 50,53,51,52,54,55,57,56,58
%N 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.
%C A088750 The zeros of the Riemann zeta function are numbered. The ordinates being 0<gamma_1 < gamma_2<gamma_3 < ... The sequence refers to the number of the zero.
%C A088750 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 terms are not true in general. The sequence is given with some mistakes in the reference arXiv:math.NT/0309433.
%C A088750 The only way I know to obtain the sequence is to draw the curves Re zeta(s)=0 and Im zeta(s)=0.
%H A088750 Juan Arias-de-Reyna, <a href="/A088750/b088750.txt">Table of n, a(n) for n = 1..79</a>
%H A088750 J. Arias-de-Reyna, <a href="https://arxiv.org/abs/math/0309433">X-Ray of Riemann zeta-function</a>, arXiv:math/0309433 [math.NT], 2003.
%H A088750 Juan Arias-de-Reyna, <a href="/A088750/a088750.pdf">Further notes on A088750</a>, September 2018.
%H A088750 A. Speiser, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002277352">Geometrisches zur Riemannschen Zetafunktion</a>, Math. Ann., Vol. 110 (1934), pp. 514-521.
%e A088750 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.
%Y A088750 Cf. A088749.
%K A088750 hard,nonn
%O A088750 1,2
%A A088750 _Juan Arias-de-Reyna_, Oct 15 2003