Editing Prime-counting function (section)

==The Riemann hypothesis==

The [[Riemann hypothesis]] implies a much tighter bound on the error in the estimate for {{math|''π''(''x'')}}, and hence to a more regular distribution of prime numbers,

:<math>\pi(x) = \operatorname{li}(x) + O(\sqrt{x} \log{x}).</math>
Specifically,<ref>{{Cite journal | last1=Schoenfeld | first1=Lowell |author-link=Lowell Schoenfeld| title=Sharper bounds for the Chebyshev functions ''θ''(''x'') and ''ψ''(''x''). II | doi=10.2307/2005976 | mr=0457374  | year=1976 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=30 | issue=134 | pages=337–360 | jstor=2005976 | publisher=American Mathematical Society}}</ref>

:<math>|\pi(x) - \operatorname{li}(x)| < \frac{\sqrt{x}}{8\pi} \, \log{x}, \quad \text{for all } x \ge 2657. </math>

{{harvtxt|Dudek|2015}} proved that the Riemann hypothesis implies that for all {{math|''x'' ≥ 2}} there is a prime {{mvar|p}} satisfying
:<math>x - \frac{4}{\pi} \sqrt{x} \log x < p \leq x.</math>