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).

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

Offset

1,4

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.

Links

Table of n, a(n) for n=1..28.

Anatolii A. Karatsuba and Ekatherina A. Karatsuba, The "problem of remainders" in theoretical physics: "physical zeta" function, 6th Mathematical Physics Meeting: Summer School and Conference on Modern Mathematical Physics, 14-23 September 2010, Belgrade, Serbia. [From Internet Archive Wayback Machine]

Tomás Oliveira e Silva, Tables of values of pi(x) and of pi2(x)

Eric Weisstein's World of Mathematics, Prime Counting Function.

Eric Weisstein's World of Mathematics, Riemann Prime Number Formula.

Eric Weisstein's World of Mathematics, Gram Series.

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 *)

Prog

(PARI) A057794=vector(#A006880, i, round(1+suminf(k=1, log(10^i)^k/(k*k!*zeta(k+1)))-A006880[i])) \\ - M. F. Hasler, Feb 26 2008

Crossrefs

Cf. A006880, A057752.

Sequence in context: A193901 A083472 A213996 * A357199 A257545 A073715

Adjacent sequences: A057791 A057792 A057793 * A057795 A057796 A057797

Keyword

sign

Author

Robert G. Wilson v, Nov 04 2000

Extensions

First term corrected by David Baugh, Nov 15 2002

Signs added by M. F. Hasler, Feb 26 2008

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

Terms a(25)-a(28) obtained using A006880. - Eduard Roure Perdices, Apr 13 2021

Status

approved