A057794 - OEIS (original) (raw)

A057794

(Integer nearest R(10^n)) - pi(10^n), where pi(x) is the number of primes <= x, R(x) = Sum_{ k>=1 } (mu(k)/k * li(x^(1/k))) and li(x) is the Cauchy principal value of the integral from 0 to x of dt/log(t).

12

1, 1, 0, -2, -5, 29, 88, 97, -79, -1828, -2318, -1476, -5773, -19200, 73218, 327052, -598255, -3501366, 23884333, -4891825, -86432204, -127132665, 1033299853, -1658989719, -1834784714, -17149335456, -17535487934, -174760519827

COMMENTS

This is Riemann's remarkable approximation for the number of primes <= x.

Equivalently, R(x) is given by the Gram series, 1 + sum of log(x)^k/(k*k!*zeta(k+1)) for k = 1 to infinity. This series converges more quickly.

REFERENCES

John H. Conway and R. K. Guy, "The Book of Numbers," Copernicus, an imprint of Springer-Verlag, NY, 1996, page 146.

M. du Sautoy, The Music of the Primes, Fourth Estate / HarperCollins, 2003; see p. 90.

MATHEMATICA

R[x_] := Sum[N[LogIntegral[x^(1/k)]*MoebiusMu[k]/k, 36], {k, 1, 1000}]; a[n_] := Abs[Round[R[10^n]-PrimePi[10^n]]]

gram[x_] := 1+Sum[N[Log[x]^k/(k*k!*Zeta[k+1]), 100], {k, 1, 1000}]; a[n_] := Abs[Round[gram[10^n]-PrimePi[10^n]]]

(* From version 7 on : *) a[n_] := Round[RiemannR[10^n]-PrimePi[10^n]] (* Jean-François Alcover, Sep 17 2012 *)

EXTENSIONS

First term corrected by David Baugh, Nov 15 2002

The value of a(23) is not known at present, I believe. - N. J. A. Sloane, Mar 17 2008

Last two terms a(23) and a(24), with Pi(10^n) for n=23 and 24 from A006880, from Vladimir Pletser, Feb 27 2013