The prime-counting function and its analytic approximations

Abstract

The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral li(The results show that π(x) < li(x) for 2 ≤ x ≤ 10 14 , and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly |π(x) − li(x)| < x 1/2 and − 2 5 x 3/2 < x 2 (π(u) − li(u))du < 0 for all x > 2. The paper concludes with a short discussion of prospects for further computational progress.

On a Constant Related to the Prime Counting Function

“…Thus, recalling the estimate π (x) ≤ Li (x) valid for 2 ≤ x ≤ 10 14 by [11] and a direct computation for small values the Lemma follows.…”

Nearest-neighbor-spacing distribution of prime numbers and quantum chaos

“….. If the Riemann Hypothesis is true then for all nontrivial zeros (ρ) = 1 2 and the contribution to the sum over k in (22) is dominated by the first term, what leads to the following approximation to π(x): [49], however on average the behavior of both differences π(x) − Li(x) and π(x) − R(x) seems to be the same [50]. The above function R(x) can be obtained, without the need of calculating the logarithmical integral Li(x), from the series obtained by J.P. Gram, see e.g.…”

A NEW BOUND FOR THE SMALLEST x WITH π(x) > li(x)

“…The smallest value of x with π(x) ≥ li(x) will be denoted Ξ, as in the recent paper by Kotnik [5]. In the course of a systematic computational study, Kotnik proves that 10 14 < Ξ.…”