The book "A SOURCE BOOK IN MATHEMATICS" has a great collection of mathematical papers. On of the is Chebyshev's Memoir on "The Totality of Primes Less Than a Given Number."
The book states that
Chebyshev did not reach the final goal - to prove that the ratio of $\phi(x):\dfrac{x}{\log x}$ tends to $1$ as $x \to \infty$.
However, in the Memoir it is presented the following:
Theorem 3. The expression $\frac{x}{{\phi \left( x \right)}} - \log x$ can not have a limit disinct from $-1$ as $x \to \infty$.
$\phi \left( x \right) \sim \frac{x}{{\log x - 1}}$
This a much better estimate. Can't it be used to show:
$\phi \left( x \right) \sim \frac{x}{{\log x}}?$
Moreover, Chevyshev proves:
Theorem 2. The function $\phi(x)$ which designates the totality of primes less than $x$, satisfies infinitely many times, between $x=2$ and $x=\infty$, each of the inequalities,
\eqalign{ & \phi \left( x \right) > \int\limits_2^x {\frac{{dx}}{{\log x}}} - \frac{{\alpha x}}{{{{\log }^n}x}} \cr & \phi \left( x \right) < \int\limits_2^x {\frac{{dx}}{{\log x}}} + \frac{{\alpha x}}{{{{\log }^n}x}} \cr}
Which gives an even better and modern estimate, which has $\dfrac{x}{{\log x}}$ as a leading term.
Why is it that the statement is given even though his results seem much greater in hierarchy?