number theory - Chebyshev/Tchebycheff's results concerning $phi(x)$.

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?

Answer

Let $\pi(x)$ be the usual prime-counting function. The first result that you mention is easly shown to be equivalent to the statement that if

$$\lim_{x\to\infty}\frac{\pi(x)}{\frac{x}{\log x}}$$
exists, then that limit must be $1$. There remained the very hard problem of proving that the limit exists. Although considerable effort was expended on the problem, the final proof only came almost a half-century later.

The second result says that the inequalities are satisfied infinitely many times, meaning that there exist arbitrarily large $x$ for which the inequalities hold. But the second result does not say that the inequalities always hold, or even that they hold for all large enough $x$.