Timeline for What is the simplest proof that the density of primes goes to zero?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 18, 2022 at 8:38 | comment | added | Roland Bacher | I guess Erdos argument boils down to $\lim \log(4)/\log(n)=0$? (Using $B_n={2n\choose n}\leq 4^n$ and $B_n$ contains all primes in $\{n+1,\ldots,2n\}$). | |
| Oct 17, 2022 at 21:12 | answer | added | Salvo Tringali | timeline score: 2 | |
| Jan 28, 2021 at 17:30 | comment | added | Esteban Crespi | There is a very simple proof in Remark on pi(x) = o(x). Proc. Amer. Math. Soc. (1962) p. 664-665 using just $\prod(1-1/p)\to 0$ and the multiplicativity of Euler's $\varphi$ function. Very similar to the proofs below. | |
| Jan 28, 2021 at 6:42 | answer | added | José Hdz. Stgo. | timeline score: 10 | |
| Jan 19, 2021 at 8:39 | comment | added | Qiaochu Yuan | @Carl: it's the same. The density of composites is bounded from below by $1 - \prod_{p \le n} \left( 1 - \frac{1}{p} \right)$ where the product is over primes and then you take $n \to \infty$, but this is just $1$ minus the same density for primes. | |
| Jan 18, 2021 at 14:59 | vote | accept | Kim | ||
| Jan 18, 2021 at 13:28 | comment | added | Carl Witthoft | Just musing - does anyone prove the complement, i.e. that the density of nonprimes approaches 100% ? | |
| Jan 18, 2021 at 0:33 | history | became hot network question | |||
| Jan 17, 2021 at 21:25 | comment | added | Martin Sleziak | Mathematics: Percentage of primes among the natural numbers, Specifically, Pete L. Clark's answer gives a proof which basically shows that $\liminf\frac{\varphi(n)}n=0$. And the details are given in robjohn's answer. (The OP accepted my answer, but Pete L. Clark's answer is definitely better.) | |
| Jan 17, 2021 at 19:27 | answer | added | Yuval Peres | timeline score: 22 | |
| Jan 17, 2021 at 19:25 | comment | added | Ofir Gorodetsky | Let me mention the Eratosthenes–Legendre Sieve. It is less involved than Erdős' argument, but more involved than the answers given below. It tells us that the density of primes is O(1/log log x); it is explained e.g. in this blog post - jonismathnotes.blogspot.com/2014/09/… . | |
| Jan 17, 2021 at 19:01 | answer | added | Fedor Petrov | timeline score: 30 | |
| Jan 17, 2021 at 17:34 | answer | added | Terry Tao | timeline score: 67 | |
| Jan 17, 2021 at 16:57 | answer | added | GH from MO | timeline score: 20 | |
| Jan 17, 2021 at 16:54 | comment | added | markvs | The ( Erdős ) proof involving $2n\choose n$ is the simplest known. | |
| Jan 17, 2021 at 16:29 | history | asked | Kim | CC BY-SA 4.0 |