Timeline for At what point would an elementary generalization of Bertrand's Postulate be interesting?
Current License: CC BY-SA 4.0
15 events
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| Feb 24, 2023 at 17:00 | history | edited | Glorfindel | CC BY-SA 4.0 |
broken link fixed, cf. https://meta.mathoverflow.net/q/5301/70594
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| Apr 8, 2016 at 13:15 | history | edited | GH from MO |
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| Apr 8, 2016 at 13:12 | answer | added | marh_g | timeline score: 2 | |
| Mar 26, 2016 at 21:51 | comment | added | john mangual | Despite being "elementary", I have eschewed such arguments in favor for proofs / write-ups that are easier to read. I am dissatisfied that I cannot map the complex analysis to elementary strategies. We are forced to accept ex nihilo | |
| Aug 30, 2013 at 23:23 | comment | added | Noam D. Elkies | I found in my lecture notes a reference to a paper by N.Costa Pereira that proves $|\psi(x)/x - 1| < 1/2976$ for $x > 10^{11}$: "Elementary estimates for the Chebyshev function psi(x) and the Möbius function M(x)", Acta Arithmetica 52 (1989), 307-337. | |
| Aug 30, 2013 at 23:17 | history | edited | user9072 |
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| Apr 12, 2013 at 10:38 | comment | added | Larry Freeman | Thanks very much for the reference! I look forward to checking it out. | |
| Apr 11, 2013 at 22:56 | comment | added | Terry Tao | Ah, I found the reference now, in this survey of Diamond: projecteuclid.org/… . In section 9 he discusses how, for each k, there is an elementary proof of Chebyshev type of a prime between $kx$ and $(k+1)x$ for large enough x. Unfortunately, the proof that the elementary proof exists (!) itself depends on the PNT! | |
| Apr 11, 2013 at 22:25 | comment | added | Terry Tao | From the explicit formula linking primes and zeroes, the above assertion for a given $k$ is morally equivalent to the absence of a zero on the line $\{ Re(s)=1 \}$ of imaginary part $O(k)$. I vaguely recall reading some discussion in which this equivalence could be made more precise, in that such a zero-free region could be converted to an elementary Ramanujan-style result (somewhat analogously to how the non-vanishing of $L(1,\chi)$ for all $\chi$ of period $q$ can be converted to an elementary proof of Dirichlet's theorem mod $q$) but I don't remember the details. | |
| Apr 11, 2013 at 22:22 | comment | added | Terry Tao | From the Erdos-Selberg argument, it is not too difficult to see (basically by iterating the Selberg symmetry formula) that if one can get $\gg_k x/\log x$ primes between $kx$ and $(k+1)x$ for all $k$ and all sufficiently large $x$, then the prime number theorem follows from elementary means (of course, this is a somewhat vacuous statement since the entire Erdos-Selberg proof of PNT is already considered elementary, but the derivation here is simpler than that of full Erdos-Selberg). | |
| Apr 11, 2013 at 18:41 | vote | accept | Larry Freeman | ||
| Apr 11, 2013 at 15:54 | answer | added | user9072 | timeline score: 20 | |
| Apr 11, 2013 at 15:34 | vote | accept | Larry Freeman | ||
| Apr 11, 2013 at 18:41 | |||||
| Apr 11, 2013 at 14:31 | answer | added | Charles | timeline score: 5 | |
| Apr 11, 2013 at 13:58 | history | asked | Larry Freeman | CC BY-SA 3.0 |