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 A210447 Number of primes <= Im(rho_n), where rho_n is the n-th nontrivial zero of Riemann zeta function. 1

%I

%S 6,8,9,10,11,12,12,14,15,15,16,16,17,17,18,19,19,20,21,21,22,22,23,23,

%T 23,24,24,24,25,26,27,27,28,29,29,30,30,30,30,30,30,31,31,32,32,32,33,

%U 34,34,34,34,34,35,35,36,36,37,37,37,38,38,39,39,39,40,40

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

%C 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.

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

%H A. Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/zeta_tables">Tables of zeros of the Riemann zeta function</a>

%H <a href="/index/Z#zeta_function">Index entries for zeta function</a>

%e 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.

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

%Y Cf. A000720, A002410, A161914, A208436.

%K nonn

%O 1,1

%A _Omar E. Pol_, Feb 03 2013

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)