Write Dirichlet's theorem on arithmetic progression #251
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| In this chapter, we will sketch a proof of the following theorem: | ||
| \begin{theorem}[Dirichlet's theorem on arithmetic progressions] | ||
| For every positive integers $a, n > 0$ such that $\gcd(a, n) = 1$, then there are infinitely |
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should this say something like every pair of positive integers?
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just a wording issue. I don't mind either, but there's another incident where the other wording is used
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i would say 16F would be better as "for all $x_1, x_2 \in X" as well, so I'll change that now.
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Isn't that firmly in the analytic branch of number theory? I'm not complaining, I'm happy to see anything related to number theory (as I've mentioned before). Actually I'm probably baiting for more, since analytic NT is explicitly mentioned as the one topic not in napkin (Topics not in Napkin section).
Though this introduces a dependency on complex analysis which is not present for the other algebraic NT chapters.
| So what is a $L$-series? In one sentence:\footnote{In my (limited) understanding, at the time of | ||
| writing this.} |
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Very unclear for the reader that the footnote is not written by Evan. May be a good place to put your disclaimer.
Is it? Neukirch's book is named "algebraic number theory" so… side note, I remember reading somewhere "algebraic number theory doesn't mean 'number theory using algebraic methods', it means 'the theory of algebraic numbers'". I should figure out what analytic number theory is soon. |
| \end{quote} | ||
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| Surprisingly, you will see $L$-series in many other places as well --- an elliptic curve over $\QQ$ | ||
| has a $L$-series, and a modular form also have a $L$-series!\todo{really? |
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"a modular form also have" -> "a modular form also has"
| So what is a $L$-series? In one sentence:\footnote{In my (limited) understanding, at the time of | ||
| writing this.} | ||
| \begin{moral} | ||
| A $L$-series bridges between an infinite sum (complex analysis) |
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Should be an
might need a disclaimer “All evidences of incompetence is not evanchen's fault.” stuck somewhere.
I'm not 100% confident that I understand that's what L-series is supposed to do, but here we go. (A missing piece of puzzle is the functional equation. There's a MathOverflow question on that but I still can't digest it yet…)
Hopefully it is sufficiently well-motivated, though the last section gets more technical.
Will there be another chapter? I'm not sure. Maybe on Hecke L-series? (at the moment I don't know if it's worth writing about yet)