Folkmar Bornemann proved, when assuming the conjecture that all zeros of the Riemann zeta function are simple, that where runs over the non-trivial zeros of the Riemann zeta function and . The sum over non-trivial zeta zeros in the formula for describes the fluctuations of while the remaining terms give the "smooth" part of prime-counting function, so one can use as a good estimator of for x > 1. In fact, since the second term approaches 0 as , while the amplitude of the "noisy" part is heuristically about estimating by alone is just as good, and fluctuations of the distribution of primes may be clearly represented with the function Inequalities Here are some useful inequalities for (x). for x ≥ 17. The left inequality holds for x ≥ 17 and the right inequality holds for x > 1. The constant 1.25506 is to 5 decimal places, as has its maximum value at x = 113. Pierre Dusart proved in 2010: for , and for . Here are some inequalities for the nth prime, pn. The upper bound is due to Rosser (1941), the lower one to Dusart (1999): for n ≥ 6. The left inequality holds for n ≥ 2 and the right inequality holds for n ≥ 6. An approximation for the nth prime number is Ramanujan proved that the inequality holds for all sufficiently large values of . In Dusart proved (Proposition 6.6) that, for , and (Proposition 6.7) that, for , More recently, Dusart has proved (Theorem 5.1) that, for , , and that, for , The Riemann hypothesis The Riemann hypothesis implies a much tighter bound on the error in the estimate for , and hence to a more regular distribution of prime numbers, Specifically, See also Foias constant Bertrand's postulate Oppermann's conjecture Ramanujan prime References Notes External links Chris Caldwell, The Nth Prime Page at The Prime Pages. Tomás Oliveira e Silva, Tables of prime-counting functions. Analytic number theory Prime numbers Arithmetic functions