A153815 Indices of nontrivial zeros of the Riemann zeta function where the real part of zeta'(s) becomes negative.

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

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 A334095

Adjacent sequences: A153812 A153813 A153814 * A153816 A153817 A153818

Keyword

nonn

Author

Vladimir Reshetnikov, Jan 02 2009

Status

approved