Timeline for What was the first result that belongs to the field of analytic number theory?
Current License: CC BY-SA 4.0
18 events
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| Jan 21, 2021 at 4:53 | comment | added | markvs | @TimothyChow: I am glad that you understand that I think that the question is ill-posed. I think it is obvious and I have no interest in changing your opinion about it.. | |
| Jan 21, 2021 at 4:04 | comment | added | Mustafa Said | @GHfromMO Gauss also conjectured that $\pi(n)$ is asymptotic to $\int_2^n \frac{1}{\ln{t} } dt$. | |
| Jan 21, 2021 at 4:00 | comment | added | Timothy Chow | @dodd: I don't think you are making a good faith attempt to answer the question as posed, but are in effect saying that the question is a poor one. I believe that the question is a reasonable one and that one should try to answer the intended question. | |
| Jan 21, 2021 at 2:47 | comment | added | markvs | @TimothyChow: I am a number theorist and I say that $1+1=2$ is one of the most important results of analytic number theory (as well as every other part of number theory and probably every other part of mathematics and physics). | |
| Jan 20, 2021 at 16:24 | comment | added | Timothy Chow | "Does not belong" means that no number theorist would say that $1+1=2$ is a result of analytic number theory. | |
| Jan 20, 2021 at 16:22 | comment | added | Timothy Chow | Today the distinction between "analysis" and "elementary arithmetic" is no longer so clear. "Elementary arithmetic" corresponds approximately to first-order Peano arithmetic (PA), but the trouble is that a lot of what we call complex analysis can, despite initial appearances, be formalized in PA, so PA does not exactly capture the older concept of "elementary proof." | |
| Jan 20, 2021 at 16:22 | comment | added | markvs | @TimothyChow: What does it mean "does not belong". The analytic number theory would not exist if $1+1\ne 2$. | |
| Jan 20, 2021 at 16:20 | comment | added | Timothy Chow | @dodd : A century or two ago, people felt that there was a fairly clear distinction between "elementary arithmetic" and "analysis." The use of analysis to prove results in arithmetic was regarded as an unusual and surprising thing, and there was a fascination with trying to find "elementary" proofs of such results (e.g., the prime number theorem). Here the word "elementary," despite appearances, does not mean "easy" or "simple," but means "without using analysis." $1+1=2$ is a theorem of arithmetic, but is elementary and does not use analysis, so it does not belong to "analytic number theory." | |
| Jan 20, 2021 at 11:03 | comment | added | GH from MO | Gauss proved results on the average of class numbers decades before Dirichlet. | |
| Jan 20, 2021 at 9:54 | comment | added | Francesco Polizzi | If we consider Diophantine Approximation as part of Number Theory, then the approximation $\pi \simeq 22/7$ obtained by Archimedes belongs to Analytic Number Theory (exhaustion method is calculus in disguise). | |
| Jan 20, 2021 at 4:32 | comment | added | KConrad | and Dirichlet as "the true father of analytic number theory". | |
| Jan 20, 2021 at 4:32 | comment | added | KConrad | To address Dirichlet vs. Euler, a label like "first analytic number theorist" has been applied to both of them. Euler not only proved the result in the post (at a level of rigor accepted in his time), but also found the Euler product for the zeta-function, a form of the functional equation for the zeta-function and $L$-function of the nontrivial character mod $4$ (at integers), and proved results about the partition function using what we'd call $q$-products. Iwaniec and Kowalski, in the introduction of their book, describe Euler's work as "the beginning of analytic number theory" (contd.) | |
| Jan 20, 2021 at 4:05 | comment | added | KConrad | @reuns note the question is asking about the first result in analytic number theory rather than the first analytic number theorist. Fermat's little theorem can be regarded today as an early result in group theory, preceding the general development of group theory by a long time, even though Fermat would not be called a group theorist. | |
| Jan 20, 2021 at 4:00 | comment | added | KConrad | Just to be precise, Euler did not deal in such a careful way with limits as you express his result. He had no variable $s$ in the limit but used $s = 1$ throughout, so he only "proved" $\sum_p 1/p = \infty$.. He sensed that $\sum_{p \leq x} 1/p$ might grow like $\log\log x$ because he wrote $\sum_p 1/p = \log \log \infty$, but he did not work with truncated expressions either. | |
| Jan 20, 2021 at 1:56 | comment | added | reuns | The first analytic number theorist is clearly Dirichlet. | |
| Jan 20, 2021 at 1:50 | comment | added | Asvin | Euler's result is the first one I can think of! (which isn't saying much) | |
| Jan 20, 2021 at 1:38 | history | edited | Will Sawin |
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| Jan 20, 2021 at 1:35 | history | asked | Mustafa Said | CC BY-SA 4.0 |