Editing Prime-counting function (section)

===More precise estimates===
In 1899, [[Charles Jean de la Vallée Poussin|de la Vallée Poussin]] proved that
<ref>See also Theorem 23 of {{cite book |author = A. E. Ingham |author-link = Albert Ingham |title = The Distribution of Prime Numbers |date=2000 |publisher = Cambridge University Press |isbn=0-521-39789-8}}</ref>
<math display=block>\pi(x) = \operatorname{li} (x) + O \left(x e^{-a\sqrt{\log x}}\right) \quad\text{as } x \to \infty</math>
for some positive constant {{mvar|a}}. Here, {{math|''O''(...)}} is the [[big O notation|big {{mvar|O}} notation]].

More precise estimates of {{math|''π''(''x'')}} are now known. For example, in 2002, [[Kevin Ford (mathematician)|Kevin Ford]] proved that<ref name="Ford">{{cite journal |author = Kevin Ford |title=Vinogradov's Integral and Bounds for the Riemann Zeta Function |journal=Proc. London Math. Soc. |date=November 2002 |volume=85 |issue=3 |pages=565–633 |url=https://faculty.math.illinois.edu/~ford/wwwpapers/zetabd.pdf |doi=10.1112/S0024611502013655 |arxiv=1910.08209 |s2cid=121144007 }}</ref>
<math display=block>\pi(x) = \operatorname{li} (x) + O \left(x \exp \left( -0.2098(\log x)^{3/5} (\log \log x)^{-1/5} \right) \right).</math>

Mossinghoff and [[Timothy Trudgian|Trudgian]] proved<ref>{{cite journal | first1 = Michael J. | last1 = Mossinghoff  | first2 = Timothy S. | last2 = Trudgian | author2-link=Timothy Trudgian| title = Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function | journal = J. Number Theory | volume = 157 | year = 2015 | pages = 329–349 | arxiv = 1410.3926 | doi = 10.1016/J.JNT.2015.05.010| s2cid = 117968965 }}</ref> an explicit upper bound for the difference between {{math|''π''(''x'')}} and {{math|li(''x'')}}:
<math display=block>\bigl| \pi(x) - \operatorname{li}(x) \bigr| \le 0.2593 \frac{x}{(\log x)^{3/4}} \exp \left( -\sqrt{ \frac{\log x}{6.315} } \right) \quad \text{for } x \ge 229.</math>

For values of {{mvar|x}} that are not unreasonably large, {{math|li(''x'')}} is greater than {{math|''π''(''x'')}}. However, {{math|''π''(''x'') − li(''x'')}} is known to change sign infinitely many times. For a discussion of this, see [[Skewes' number]].