A210447 Number of primes <= Im(rho_n), where rho_n is the n-th nontrivial zero of Riemann zeta function.

6, 8, 9, 10, 11, 12, 12, 14, 15, 15, 16, 16, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 26, 27, 27, 28, 29, 29, 30, 30, 30, 30, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 34, 35, 35, 36, 36, 37, 37, 37, 38, 38, 39, 39, 39, 40, 40

Offset

1,1

Comments

The zeros -2, -4, -6, ... of the Riemann zeta function are considered trivial. The nontrivial zeros are in the "critical strip" 0 < Re(rho_n) < 1. All of the known nontrivial zeros have real part 1/2. In this sequence, we count the prime numbers less than or equal to the imaginary part of these nontrivial zeros.

The Riemann hypothesis (currently unproven) states that all of the nontrivial zeros have real part 1/2.

Links

Table of n, a(n) for n=1..66.

A. Odlyzko, Tables of zeros of the Riemann zeta function

Index entries for zeta function

Example

a(8) = 12 because the 8th nontrivial zero of Riemann zeta function is 0.5 + (40.91...)i and there are 12 primes less than or equal to 40.91...; they are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and 37.

Mathematica

f[n_] := PrimePi@ Im@ ZetaZero@ n; Array[f, 70] (* Robert G. Wilson v, Jan 27 2015 *)

Crossrefs

Cf. A000720, A002410, A161914, A208436.

Sequence in context: A096391 A190572 A182619 * A135558 A031951 A043610

Adjacent sequences: A210444 A210445 A210446 * A210448 A210449 A210450

Keyword

nonn

Author

Omar E. Pol, Feb 03 2013

Status

approved