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

Author

Omar E. Pol, Feb 03 2013

Keywords

• nonn

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.

Examples

			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.
		

Links

• A. Odlyzko, Tables of zeros of the Riemann zeta function
• Index entries for zeta function

Crossrefs

Cf. A000720, A002410, A161914, A208436.

Programs

Mathematica

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