# How did Gauss discover the prime number theorem?

Carl Friedrich Gauss conjectured in his early youth that

$$\lim_{x \rightarrow \infty} \frac{\pi(x)}{x/\log(x)} = 1.$$

Any idea how did he reach such result?

• Probably by counting the number of primes between $1$ and $x$ for many different values of $x$. By the way, I'd use $n$ instead of $x$ here. In any case, this looks more like an opinion-based question. If there is a concise answer to it, then it might be under Guass page on Wikipedia (or some other related source of knowledge). Jan 20 '15 at 19:24
• ...plotting the result of his counting, and making an educated guess about the growth of the resulting curve. Jan 20 '15 at 19:26
• what I read was that he counted primes in intervals, not necessarily starting at 1, and found that near some $x$ the percentage of primes was about $1 / \log x.$ Jan 20 '15 at 19:28
• Vaguely related: Does the $x/\log x$ somehow pop up naturally in the proof or does one get some other expression which is proportional to it first? (e.g. $\mathrm{Li}(x)$) Jan 20 '15 at 19:29
• Do you mean the actual historical reason or some idea how one could come up with it? For the former also consider History of Science and Mathematics site.
– quid
Jan 20 '15 at 19:31

It is not quite accurate to say that Gauss discovered the prime number theorem. He made tables of the quantity

$$\int_2^x\frac{dt}{\ln t}$$

which are reprinted in Harold Edwards' book Riemann's Zeta Function. According to Edwards citing Gauss's Werke, Vol. II Gauss claims (in an 1849 letter) to have conjectured that the density of primes was $1/\log x$ around 1792 or 1793.

According to Edwards, Gauss presented no analytic basis for the conjecture but gave empirical data in the form of tables suggesting the relation.

According to the Wiki entry on the PNT Legendre in 1797 or 1798 conjectured a relation of the form $a/(A\ln a + B)$ which he later sharpened. Chebyshev subsequently showed something akin to the prime number theorem but shy of it: that if

$$\lim_{x\to\infty}\frac{\pi(x)}{\int_2^x\frac{dt}{\ln t}}$$ exists the limit is 1.

Edwards and one of his sources say that the relation given by Legendre had been around for some time. Since Gauss didn't publish it the original source of the general idea is arguably obscure.

By the time the prime number theorem was finally proven in 1896 by Hadamard and de la Vallee Poussin (separately) is was much more than a guess and not quite a theorem.

Gauss deserves credit for the numerical observation and for the unpublished conjecture because his credibility in matters of priority is high (which Edwards and others discuss). Properly speaking we can give him credit for the prime number conjecture.

• Thanks for such detailed informations. Jan 21 '15 at 13:14
• What you mean with "not quite a theorem"? Dec 16 '18 at 1:50
• @BillyRubina: Prior to the proof it was not a theorem. It was widely believed to be true. That two people managed to prove it at the same time with somewhat different approaches suggests (to me) that analysis had matured enough to make a proof more or less inevitable. So "not quite" or "all but" a theorem. Dec 16 '18 at 7:37