Inequalities concerning the function π(x): Applications
Volume 94 / 2000
Abstract
Introduction. In this note we use the following standard notations: π(x) is the number of primes not exceeding x, while $θ(x) = ∑_{p≤x} log p$. The best known inequalities involving the function π(x) are the ones obtained in [6] by B. Rosser and L. Schoenfeld: (1) x/(log x - 1/2) < π(x) for x ≥ 67 (2) x/(log x - 3/2) > π(x) for $x > e^{3/2}$. The proof of the above inequalities is not elementary and is based on the first 25 000 zeros of the Riemann function ξ(s) obtained by D. H. Lehmer [4]. Then Rosser, Yohe and Schoenfeld announced that the first 3 500 000 zeros of ξ(s) lie on the critical line [9]. This result was followed by two papers [7], [10]; some of the inequalities they include will be used in order to obtain inequalities (11) and (12) below. In [6] it is proved that π(x) ~ x/(log x - 1). Here we will refine this expression by giving upper and lower bounds for π(x) which both behave as x/(log x - 1) as x → ∞.
