A282793 Indices k of nontrivial Riemann zeta zeros such that floor(Im(zetazero(k))/(2*Pi)*log(Im(zetazero(k))/(2*Pi*e)) + 7/8) - k + 1 = 1.

127, 196, 233, 289, 368, 380, 401, 462, 519, 568, 596, 619, 627, 655, 669, 693, 716, 729, 767, 796, 820, 849, 858, 888, 965, 996, 1029, 1035, 1044, 1114, 1179, 1210, 1251, 1277, 1291, 1308, 1332, 1343, 1431, 1457, 1488, 1496, 1499

Offset: 1

Author

Mats Granvik, Feb 21 2017

Keywords

• nonn

Comments

Conjecture 1: The union of this sequence and A282794 is A153815.

Conjecture 2: The zeta zeros whose indices are terms of this sequence are the locations where the zeta zero counting function, (RiemannSiegelTheta(t) + Im(log(zeta(1/2 + i*t))))/Pi + 1 overcounts the number of zeta zeros on the critical line.

Conjecture 3: This sequence consists of the numbers k such that sign(Im(zetazero(k)) - 2*Pi*e*exp(LambertW((k - 7/8)/e))) = 1. Verified for the first 100000 zeta zeros.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Mathematica

(* Definition: *)
fQ[n_] := Block[{a = N[Im@ ZetaZero@ n, 32]}, Floor[a (Log[a] - Log[2Pi] - 1)/(2Pi) + 7/8] == n]; Select[ Range@ 1550, fQ] (* Robert G. Wilson v, Feb 21 2017 *)
(* Definition: *)
Monitor[Flatten[Position[Table[Floor[Im[ZetaZero[n]]/(2*Pi)*Log[Im[ZetaZero[n]]/(2*Pi*Exp[1])] + 7/8] - n + 1, {n, 1, 1500}], 1]], n]
(* Conjecture 3: *)
Monitor[Flatten[Position[Table[Sign[Im[ZetaZero[n]] - 2*Pi*E*Exp[LambertW[(n - 7/8)/E]]], {n, 1, 1500}], 1]], n]