New fast methods to compute the number of primes less than a given value

UDC 519.688 The paper describes new fast algorithms for evaluating π ( x ) inspired by the harmonic and geometric mean integrals that can be used on any pocket calculator.  In particular, the formula h ( x ) based on the harmonic mean is within ≈ 15 of the actual value for 3 ≤ x ≤ 10000.  The approximation verifies the inequality, h ( x ) ≤ L i ( x ) and, therefore, is better than L i ( x ) for small x .   We show that h ( x ) and their extensions are more accurate than other famous approximations, such as Locker–Ernst's or Legendre's also for large x .   In addition, we derive another function g ( x ) based on the geometric mean integral that employs h ( x ) as an input, and allows one to significantly improve the quality of this method.  We show that g ( x ) is within ≈ 25 of the actual value for x ≤ 50000 (to compare L i ( x ) lies within ≈ 40 for the same range) and asymptotically g ( x ) ∼ x ln x exp ( 1 ln x - 1 ) .