We "lose" a year from the weird introductory year, then.
@anon " In it he used a translation invariant integration on the locally compact group of ideles to lift the zeta function of a number field, twisted by a Hecke character, to a zeta integral and study its properties." LOL
Do you understand where algebraic geometry came from? Do you understand what it's an analogy of? For example, do you know why locally free, locally free O_X-modules are important?
depending on how big it is it can get pretty hard. usually i'd start by looking for direct factors, then identify its characteristic subgroups (center, fitting, frattini commutator)
then i'd take chief series of each sylow subgroups if it's solvable, otherwise first i'd look at a composition series
there's like a bajillion $p$-groups though, most of the time it's not really worth it to completely determine the isomorphism class.
@AlexYoucis I was pretty amazed when I was given the definition of a derivative of a multivariable function in the Euclidean space as a map from $\Bbb R^n$ to $\operatorname{hom}(\Bbb R^n,\Bbb R^m)$! Although Rudin's previous discussion made it more natural.
@AlexYoucis there's a paper of hillar and rhae which has an algorithmic method for finding automorphism groups of finite abelian groups, it gets pretty nast-ay.
the entire book is a single wall of text with barely any paragraph breaks or sectioning, and every theorem has like 35 hypothesis. oh and it's in typewriter.
i still lack a lot of motivation for the subject though, a lot. and i wish i didn't - even at the most basic sense i still struggle to care about complex functions at all, but other people certainly seem to like them.
@AlexanderGruber I mean, at least from a naive point of view, it's a natural question to ask. We want to do calculus on R^2, but since R^2 happens to come with a nice field operation what happens if we try to do "one-variable R analysis" in C, formally replacing all x's with z's
why it's interesting is a different question though
you'll be happy to hear that complex analysis is, in some sense, the most algebraic analysis
@AlexYoucis there may be something rebellious in my spirit, but a lot of what seems interesting to me in complex analysis so far seems to be things others want to avoid. i was excited to study non-rectifiable curves, but most of the chapter spent time talking about how to avoid them.
@robjohn I have been trying to write my own proof of dirichlets theorem with out characers, I was able to show if, $$\lim_{s\to 1}(s-1)\sum_{n=0}^\infty\frac{\mu(an+b)}{(an+b)^s}=0$$, then the primes in coprime residue classes have dirichlet density $\frac{1}{\phi(a)}$, I eventually found a way to solve for $$\sum_{n=0}^\infty\frac{\mu(an+b)}{(an+b)^s}$$ in terms of a system of equations, and I could show it was $O(1)$ so long as the system was solveable
But the eigen values turned out to be precisely the dirichlet L functions with characters modulo a
I got this, though $$\sum_{p \equiv b \text{ mod a}}\frac{\ln(p)^2}{p^s}+\sum_{pq \equiv b \text{ mod a}}_{p\ne q}\frac{\ln(p)\ln(q)}{(pq)^s}=\frac{1}{\phi(a)}\frac{1}{(s-1)^2}+O(\frac{1}{(s-1)})$$
sofar by elementry means
So I can show, atleast half of the coprime arithimetic progressions contain infinitely many primes
Its remarkably easier to obtain results on semi primes, then it is primes
Change of square functions are quite tasty results.
@robjohn It is time you found functional calculi interesting! So then you take an unbounded operator and the analytic function on a sector $z \mapsto \sqrt{z} \exp(z)$ and plug in $L$!
And then you are like holy cow, it is a bounded operator.
someone flaged an answer as not an answer and i flaged it with invalid flag, and my "invalid flag" was disputed with the comment "disputed - flags should not be used to indicate technical inaccuracies, or an altogether wrong answer"
