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 $\log x$.

  • $\begingroup$ Which follows easily from $\log \zeta(s) = s \int_1^\infty \text{Li}(x) x^{-s-1}dx +\mathcal{O}(1)$ as $s \to 1^+$ $\endgroup$ – reuns Aug 29 '17 at 23:17

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