A215663 Floor(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).

0, 0, 0, -3, -5, 29, 88, 96, -79, -1828, -2319, -1476, -5774, -19201, 73217, 327052, -598255, -3501366, 23884333, -4891825, -86432205, -127132665, 1033299853, -1658989720, -1834784715, -17149335456, -17535487935, -174760519828

Offset: 1

Author

Vladimir Pletser, Mar 09 2013

Keywords

• sign

Comments

In Riemann's approximation for the number of primes <= 10^n, taking Floor(R(10^n)), i.e. the greatest integer <= R(10^n), instead of the nearest integer to R(10^n), i.e. Round(R(10^n)) (see A057794), provides a better approximation to pi(10^n) for small values of n and some other values of n, i.e. Abs(a(n)) = Abs(A057794(n))-1 for n = 1, 2, 8, 15. However, the approximation is worse by one unit, i.e. Abs(a(n)) = Abs(A057794(n))+1 for n = 4, 11, 13, 14, 21, 24, 25, 27, 28. The approximation is the same for the other 15 values of n <= 28. However, it yields a better average relative difference, i.e. Average(Abs(a(n))/pi(10^n)) = 1.24535…x10^-4 for 1 <= n <= 28, compared to Average(Abs(A057794(n))/pi(10^n)) = 1.04526…x10^-2. - Corrected and extended by Eduard Roure Perdices, Apr 16 2021

References

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

Links

• Michel Planat and Patrick Solé, Improving Riemann prime counting, arXiv:1410.1083 [math.NT], 2014.

Crossrefs

Cf. A006880, A057752, A057794.

Programs

Mathematica

R[x_] := Sum[N[LogIntegral[x^(1/k)]*MoebiusMu[k]/k, 36], {k, 1, 1000}]; a[n_] := Floor[R[10^n]-PrimePi[10^n]]
a[n_] := Floor[RiemannR[10^n] - PrimePi[10^n]] (* Eduard Roure Perdices, Apr 16 2021 *)

Extensions

a(17) corrected, a(25)-a(28) obtained using A006880. - Eduard Roure Perdices, Apr 16 2021