# The first proof (Tchebychef's) regarding the asymptotic distribution of primes

Quoting from the first page of the survey article of prof. D. Goldfeld on the elementary proof of the Prime Number Theorem (PNT):

The first paper in which something was proved at all regarding the asymptotic distribution of primes was Tchebychef’s first memoir ([Tch1]) which was read before the Imperial Academy of St. Petersburg in 1848. In that paper Tchebychef proved that if any approximation to $$\pi(x)$$ held to order $$x/\log(x)^N$$ (with some fixed large positive integer $$N$$) then that approximation had to be $$\text{Li}(x)$$. It followed from this that Legendre’s conjecture that $$\lim_{x \to \infty} A(x) = 1.08366$$ was false, and that if the limit existed it had to be 1.

Here $$\pi(x)$$ is the prime counting function and the equation Legendre’s conjecture relates to is (I think?) $$\pi(x) = x/(\log(x) - A(x))$$; $$\text{Li}(x)$$ refers to the offset logarithmic integral.

I am confused on two points:

• What did Tchebychef prove exactly?

• How does this disprove Legendre's conjecture?

Here is a slightly more detailed description:

Chebyshev proved that for any $\epsilon > 0$ and any positive integer $m$, each of the inequalities $\pi(x) - \text{Li}(x) < \epsilon x(\log x)^{-m}$ and $\pi(x) - \text{Li}(x) > -\epsilon x(\log x)^{-m}$ holds for infinitely many $x$.
His methods were not strong enough to prove convergence (that would require both inequalities to hold simultaneously for all sufficiently large $x$), but if we assume that $\pi(x)$ admits an asymptotic expansion of the form $$\pi(x) = c_1 \frac{x}{\log x} + c_2 \frac{x}{(\log x)^2} + c_3 \frac{x}{(\log x)^3} + \cdots,$$
then Chebyshev's result shows that the only possible values for $c_1,c_2,\ldots$ are the ones that arise from the asymptotic expansion of $\text{Li}(x)$, namely $c_k = (k-1)!$:
$$\text{Li}(x) = \frac{x}{\log x} + \frac{x}{(\log x)^2} + 2\frac{x}{(\log x)^3} + 6\frac{x}{(\log x)^4} + \cdots$$
Looking at the first two terms of this expansion, it is easy to extract that this gives conditionally $A(x) \to 1$. While it doesn't prove that $A(x) \to 1$, it does prove that $A(x) < 1 + \epsilon$ infinitely often, hence $A(x) \not \to 1.08366$. Legendre had suggested $x/(\log x - 1.08366)$ would be a more accurate asymptotic than just $x/\log x$, which is nominally true but it only wins by a constant factor, whereas we now know that $x/(\log x - 1)$ is more accurate by a factor of $\log x$.
• Which follows easily from $\log \zeta(s) = s \int_1^\infty \text{Li}(x) x^{-s-1}dx +\mathcal{O}(1)$ as $s \to 1^+$ – reuns Aug 29 '17 at 23:17