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One of the most important theorems proved in the 19th century is the prime number theorem. Put $\pi(x)$ for the number of prime numbers $p$ satisfying $1 \leq p \leq x$. Then the prime number theorem states that

$$\displaystyle \pi(x) \sim \frac{x}{\log x}.$$

Prior to the proof of the prime number theorem, Chebyshev had proved using much more elementary arguments that there exist constants $c_1, c_2 > 0$ such that

$$\displaystyle \frac{c_1 x}{\log x} < \pi(x) < \frac{c_2 x}{\log x}$$

for $x$ sufficiently large.

What are some explicit results of the form

$$\displaystyle \frac{x}{\log x} + \frac{c_1 x}{(\log x)^2} < \pi(x) < \frac{x}{\log x} + \frac{c_2 x}{(\log x)^2}$$

for all $x > c_3$, where $c_1, c_2, c_3$ are all explicit constants?

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    $\begingroup$ In 1998, Pierre Dusart proved in his PhD thesis (in French) that $f(x) < \pi(x) < f(x) + 2.51 x/(\log x)^3$ for $x \ge 35.6 \cdot 10^4$, where $f(x) = x/\log x + x /(\log x)^2$. $\endgroup$
    – Salvo Tringali
    Commented Jun 27, 2018 at 20:02
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    $\begingroup$ In addition to Dusart, there is the 1962 work of Rosser and Schoenfeld. Likely anything recent of interest on this subject references one of those two works. Gerhard "Cited Them More Than Once" Paseman, 2018.06.27. $\endgroup$
    – Gerhard Paseman
    Commented Jun 27, 2018 at 20:27

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Rosser and Schoenfeld in their 1962 paper published in the Illinois Journal of Mathematics have as their Theorem 1:

$$ \frac{x}{\log x}(1 + \frac{1}{2\log x}) \lt \pi(x) \textrm{ for } 59 \leq x$$

$$ \pi(x) \lt \frac{x}{\log x}(1 + \frac{3}{2\log x}) \textrm{ for } 1 \lt x$$

Gerhard "Finds Comfort In The Classics" Paseman, 2018.06.27.

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