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A153815 Indices of nontrivial zeros of the Riemann zeta function where the real part of zeta'(s) becomes negative. 8
127, 136, 196, 213, 233, 256, 289, 368, 379, 380, 399, 401, 462, 509, 519, 531, 568, 580, 596, 619, 627, 639, 655, 669, 693, 696, 705, 716, 729, 767, 779, 795, 796, 809, 820, 849, 858, 871, 888, 965, 994, 996 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

From Mats Granvik, Feb 21 2017: (Start)

Conjecture 1: Indices n of nontrivial zeros of the Riemann zeta function such that: abs(floor(im(zetazero(n))/(2*Pi)*log(im(zetazero(n))/(2*Pi*e)) + 7/8) - n + 1) = 1.

Conjecture 2: The zeta zeros with these indices are also the locations where the zeta zero counting sequence A135297 disagrees with the zeta zero counting function: (RiemannSiegelTheta(t) + im(log(zeta(1/2 + I*t))))/Pi + 1. The locations where the counting function overcounts are given by A282793, and the locations where the counting function undercounts are given by A282794.

Conjecture 3: Union of A282793 and A282794.

(End)

Floor(im(zetazero(n))/(2*Pi)*log(im(zetazero(n))/(2*Pi*e)) + 7/8) - n + 1 is the branch of the argument of zeta at the n-th zero on the critical line (conjectured). - Stephen Crowley, Mar 09 2017

LINKS

Table of n, a(n) for n=1..42.

Stephen Crowley, An Expression For The Argument of zeta at Zeros on the Critical Line, arXiv:1703.03490 [math.NT], 2017.

EXAMPLE

Re(zeta'(zetazero(127))) < 0.

MATHEMATICA

Select[Range[1000], N[Re[Zeta'[ZetaZero[ # ]]] < 0] &]

(* Conjecture 1: *) Monitor[Flatten[Position[Table[Abs[Floor[Im[ZetaZero[n]]/(2*Pi)*Log[Im[ZetaZero[n]]/(2*Pi*Exp[1])] + 7/8] - n + 1], {n, 1, 1000}], 1]], n] (* Mats Granvik, Feb 21 2017 *)

CROSSREFS

Cf. A002505, A135297, A153815, A273061, A282793, A282794, A282896, A282897.

Sequence in context: A075595 A133781 A255227 * A194634 A126096 A164966

Adjacent sequences:  A153812 A153813 A153814 * A153816 A153817 A153818

KEYWORD

nonn

AUTHOR

Vladimir Reshetnikov, Jan 02 2009

STATUS

approved

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Last modified February 18 20:25 EST 2019. Contains 320262 sequences. (Running on oeis4.)