# 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?

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The first result you mention is that if the limit exists, it is what we now know to be true. There remained the very hard problem of proving that the limit exists, which only came almost a half-century later. And the second result says infinitely many times, it does not say not always. –  André Nicolas Apr 12 '12 at 0:07
I guess that can be accepted as an answer, @André. Thanks. –  Pedro Tamaroff Apr 12 '12 at 0:13

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 $x$.