Timeline for What is the high-concept explanation on why real numbers are useful in number theory?
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19 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2019 at 9:44 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Apr 14, 2011 at 17:38 | comment | added | user9072 | +1 (and much more if I could) to Mark Meckes. The question what one can or cannot do in principle (sort-of reverse mathematics) and what could be useful to progress on (or at least a substitute for existing tools for) the number theoretical questions in practice seem quite othogognal to me. And I think the question at least also referred to the latter. (In order to avoid a misunderstanding, I do not want to dismiss the former considerations, but they just seem to be on something else; and I am also aware that some logic/model theory arguments are of practical relevance in NT.) | |
| Apr 14, 2011 at 16:51 | comment | added | Mark Meckes | Yes, "can't" is certainly an overstatement. The real (no pun intended) -- and increasingly less mathematical -- question is whether such statements can be made without appealing to the reals and still be comprehensible to human beings. | |
| Apr 14, 2011 at 16:30 | comment | added | Timothy Chow | I agree with Todd; it's far from clear that there is any version of the PNT that cannot be stated without using the real numbers. Usually statements involving the growth rate of explicitly defined real functions can be converted into statements involving only rational numbers, using standard tricks familiar to logicians. Ugly and inelegant, of course, but not impossible. | |
| Apr 14, 2011 at 16:01 | comment | added | Todd Trimble | Precise versions that concern the order of growth of, e.g., $\pi(x)$ - \text{Li}(x)$, are obviously most easily stated by appeal to functions of a real variable. Whether they can't be stated without appealing to the reals is far less clear to me! Could be an interesting question... | |
| Apr 14, 2011 at 15:27 | comment | added | Mark Meckes | @Todd: Fair enough. Of course it depends on which statement one means by PNT. How about: "The most precise known versions of the Prime Number Theorem can't even be stated without real numbers." | |
| Apr 14, 2011 at 14:55 | comment | added | Todd Trimble | @Mark: I was merely questioning Gerry's statement that PNT couldn't be stated without the reals. I'm not asserting or even suggesting anything about Daniel's larger question. | |
| Apr 14, 2011 at 13:32 | comment | added | Mark Meckes | @Daniel and @Todd: Of course even the $x/\log(x)$ estimate in PNT is just an approximation to the better asymptotic $\pi(x) \sim \operatorname{Li}(x)$, which is even more difficult to discuss without real numbers. | |
| Apr 14, 2011 at 12:35 | comment | added | Cam McLeman | @Daniel: Without a metric for bestness, I don't think it's particularly reasonable to ask for a proof that any given proof is best. Of course you can argue that a better one may be found, and no one could possibly be able to contradict you. | |
| Apr 14, 2011 at 12:32 | comment | added | Cam McLeman | +1: From my viewpoint, both Erdos-Selberg and C-D have a long way to go before besting any of the standard complex-analytic proofs. | |
| Apr 14, 2011 at 11:33 | comment | added | Todd Trimble | Can't you asymptotically replace $\log(n)$ by harmonic numbers, which can be defined in the rationals? So that stating PNT doesn't require the reals? | |
| Apr 14, 2011 at 11:03 | comment | added | user9072 | @Gerry Myerson: now we only have to hope nobody proved a version of the second result replacing the series by finite sums and $\pi$ by rational approximations to it depending on the length of the sum :) More seriously, +1 for in my opinion very convincing examples. | |
| Apr 14, 2011 at 5:31 | history | edited | Gerry Myerson | CC BY-SA 3.0 |
added 129 characters in body
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| Apr 14, 2011 at 5:29 | comment | added | Gerry Myerson | @Daniel, I'm going to edit my answer to give you another target. | |
| Apr 14, 2011 at 5:27 | comment | added | Gerry Myerson | @Daniel, I'll take your word for it. But you asked about "best possible" proofs, and whether introducing real numbers is useful. Do you think the C-D proof is better than the classical ones using complex analysis (or the Erdos-Selberg, using real analysis)? It may be possible to prove (something equivalent to) PNT without reals, but history suggests introducing the reals was useful. | |
| Apr 14, 2011 at 4:43 | comment | added | Daniel Moskovich | If I'm understanding the paper I linked to, Cornaros-Dimitracopoulos replace the logarithm by the "approximate logarithm" which is an integer, and limits and infinite sums by bounded induction. So the statement and proof they give is completely over the integers (thanks Wikipedia!). | |
| Apr 14, 2011 at 4:37 | comment | added | Gerry Myerson | The Erdos-Selberg proof avoided the use of complex numbers - you asked about avoiding the use of real numbers, which Erdos-Selberg sure don't do. I don't know about Comaros-Dimitracopoulos, but the PNT says the number of primes up to $x$ is asymptotic to $x/\log x$, and I don't know how to make any sense of the logarithm without leaving the rationals for the reals. | |
| Apr 14, 2011 at 4:34 | comment | added | Daniel Moskovich | I don't understand this answer- didn't Cornaros and Dimitracopoulos formulate and exhibit a proof of the Prime Number Theorem in $I\Delta_0+(\exp)$, which is weaker than Peano Arithmetic: mpla.math.uoa.gr/~cdimitr/files/publications/… And what of the Erdos-Selberg proof? | |
| Apr 14, 2011 at 4:08 | history | answered | Gerry Myerson | CC BY-SA 3.0 |