On the estimates of the upper and lower bounds of Ramanujan primes

摘要

For n >= 1, the nth Ramanujan prime is defined as the least positive integer R-n such that for all x >= R-n , the interval (x/2, x) has at least n primes. Let p(i) be the ith prime and R-n = p(s) . Sondow, Laishram, and other scholars gave a series of upper bounds of s. In this paper we establish several results giving estimates of upper and lower bounds of Ramanujan primes. Using these estimates, we discuss a conjecture on Ramanujan primes of Sondow-Nicholson-Noe and prove that if n > 10(300), then pi (R-mn) <= m pi (R-n) for m >= 1.