Timeline for Motivation behind Analytic Number Theory
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| when toggle format | what | by | license | comment | |
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| Feb 15, 2019 at 20:52 | comment | added | Sylvain JULIEN | I'm extremely far from being a specialist like GH but I think some algebraic concepts could help make ANT even more powerful. But it seems very few people feel at ease with both algebra and analysis. | |
| Feb 15, 2019 at 16:33 | answer | added | Timothy Chow | timeline score: 27 | |
| Feb 15, 2019 at 2:45 | history | edited | GH from MO |
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| Feb 15, 2019 at 2:36 | comment | added | GH from MO | I think that the "overarching idea behind the study of analytic number theory" is that analytic methods are highly effective in answering questions about numbers. If you like analysis and numbers, then it is for you, otherwise it is not. BTW Langlands famously said that "analytic number theory lacks concepts". Well, I don't know. I think $L$-function is a great concept. | |
| Feb 15, 2019 at 1:45 | comment | added | reuns | What you are asking applies to any $\ge$ graduate level in maths, aka. pure maths, with very few concrete applications outside of maths. Taken alone the fields are just curiosities. But put together, they lead to many developments of the undergraduate maths (theory of : distributions, representations of groups, elliptic curves, number fields, $p$-adic numbers, manifolds, algebraic varieties, logic theories, L-functions, modular forms..) | |
| Feb 15, 2019 at 1:40 | comment | added | user135845 | @reuns I agree that Davenport is quite clear, but your comment doesn't really address the question as far as I see it. I was looking for something more along the lines of where such techniques get employed, outside of the confines of the book itself. If the only utility of an esoteric technique is to prove a theorem, then, the technique in itself, as I see it, is not of very great value, once the said theorem has been proved. | |
| Feb 15, 2019 at 1:20 | comment | added | user135845 | @WhatsUp That makes perfect sense. I shall ensure I do not abandon the method completely. Thanks. | |
| Feb 15, 2019 at 1:20 | comment | added | user135845 | @paulgarrett Thank you for the reply regarding my over-interpretation of the Moore-method ideas. I found your succinct explanation of the strategy one must employ while reading material in general quite educational. | |
| Feb 15, 2019 at 0:48 | comment | added | reuns | Davenport's book is quite clear in that it introduces the theory of L-functions and aims at proving the prime number theorem and in arithmetic progressions. It finishes with a few more problems and results in the field. Whose development leads to a wide range of deep problems at the boundary of many different fields (the automorphic Langland program things) the latter being fun because it needs much more than the undergraduate level in analysis, algebra, arithmetic, geometry, groups. ANT = solving arithmetic problems (most time involving the primes) using tools in analysis | |
| Feb 15, 2019 at 0:03 | comment | added | paul garrett | As @WhatsUp commented, indeed, "attempting and failing" is almost universally hugely informative (not to mention modesty/humility-generating). A relatively minor issue is about how much time to allocate to a probably-failing venture before declaring it operationally hopeless. A non-trivial question. | |
| Feb 14, 2019 at 23:50 | answer | added | Gerhard Paseman | timeline score: 11 | |
| Feb 14, 2019 at 23:38 | comment | added | WhatsUp | In complementary to what @paulgarrett said, I think you should interpret that sentence ("attempting to prove it by myself") as stressed on the word "attempting". It's impossible (and pointless) to prove everything by yourself, but "attempting" to do so will give you a much better idea of where the difficulty of the problem lies, hence a deeper understanding when you read the proof. | |
| Feb 14, 2019 at 23:33 | comment | added | paul garrett | ... [cont'd] we should admit that many other very smart, hard-working people have come up with many great ideas, which we'd only eventually experimentally discover after centuries of trial-and-error, if then. Sure, let's not discourage ourselves by thinking that the big-shots of the past did everything... but we should respect not only the historically-notable, but also many other, people just as smart/talented as we are who worked hard their whole lives to understand things. | |
| Feb 14, 2019 at 23:32 | comment | added | paul garrett | A quick comment, in lieu of a possibly better answer, etc: in my opinion, you are quite correct in perceiving/feeling/thinking that it would be extremely difficult (impossible, really) to "prove everything for yourself" in this field. For that matter, all my experience indicates that it is equally ridiculous to imagine that novices would be able to prove for themselves all the major results in any field at all. Whoa, yes, the problem is an over-interpretation of Moore-method ideas: sure, it's good to think about thing oneself... but [cont'd] | |
| Feb 14, 2019 at 23:30 | review | First posts | |||
| Feb 15, 2019 at 5:21 | |||||
| Feb 14, 2019 at 23:25 | history | asked | user135845 | CC BY-SA 4.0 |