@robjohn actually this is what I did for $6$ version! (surely, to perform a differentiation to both sides of the identity make it definitely useful). I used that when I worked on the infinite triple series version.
@DanielRust "The word "sine" comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha"?
"One of the most surprising proofs of the quadratic reciprocity law is Eisenstein's proof using the sine function. Replacing trigonometric functions (Z-periodic holomorphic functions) by elliptic functions (say Z[i]-periodic meromorphic functions), the quartic reciprocity law follows just as easily. We also discuss proofs of the quadratic reciprocity laws in quadratic number fields and mention in passing a connection to Kronecker's Jugendtraum."
@FernandoMartin Si. Estudia NT no?7 El otro dia nos dio un crash course en espacios cocientes para hablar del Teorema de la Dimension "correctamente". Un grande.
Pero no sé, al menos en mi cursada de álgebra II nunca reinterpretamos los teoremas de lineal en términos de cocientes. Me parece que es algo que en algún momento se debería hacer.
a posteriori if one has established something with facts that one has come to have access to over time or through observation or whatever. a priori if one can assume or conclude something from the get-go. a fortiori I had to look up, it means one is arguing X from claim(s) which are much stronger than X
@AnthonyCarapetis "A compact space has property $X$, a fortiori a locally compact space has property $X$" is correct, while I don't believe "$P(n)$ for all $n$, a fortiori $P(1)$" would be correct.
@AnthonyCarapetis You're right, I needed to state it differently: "P(n) for all n, a fortiori P(1) because P(n) when n=1" is not a correct use of a fortiori; the term is used when an argument for the latter proceeds in a totally analogous way as the argument for the former. Instantiation involves an explicit use of the proposition, so it would be wrong to say that you are arguing a fortiori, unless your original argument was circular. :)
Ademas, siempre te tiene picando, te hace notar que hay boludeces que quiza tendrías que saber, o te habla de "variedad", "espacio tangente" y todo eso para que te enganches.
@TedShifrin I'd like to ask you a serious question if you don't mind. How much do you feel as though having a famous advisor helped you get a faculty position?
No question the adviser's connections are important. But ultimately the quality of one's work, publications are huge. And the adviser writes one letter. For promotion, down the road, often the adviser is banned.
He says in theorem 149 (what I just said) let me set $\chi(p)=e^{\nu i}$, $\eta =p^{-s}$. He doesn't specify a rule for $\chi(p)$; but I guess he just write $e^{\nu i}$ because we know the number is a root of unity, yes? I mean, the non specification is irrelevant.
Landau makes it nice. Apostol's exposition, different.
Landau is more concise, Apostol rightfully dwells into significant details, but some months ago that'd have made me dizzy. His proof of Dirichlet's theorem has like 8 lemmas, and Landau has slicker moves. He knows his shit.
$1$ if $\chi(p)=0$ $(1+0+\cdots+0)$, is $1+1+1+\cdots \geqslant 1$ if $\chi(p)=1$ and if $\chi(p)=-1$, its $0$ if $2\not\mid l$, $1$ if $2\mid l$, @anon. Then Landau says "from the above $$f(a)\geqslant \begin{cases} 0&\text{ always } \\&\;, \text{ for quadratic } a\end{cases}$$
@robjohn Using mod(n) approach, we get that $$-1-\frac{1}{9}+ \sum_{k=0}^{\infty} \left(\frac{1}{(10 k - 3)^2} + \frac{1}{(10 k - 1)^2} + \frac{1}{(10 k + 1)^2} + \frac{1}{(10 k + 3)^2}\right)=\frac{3\pi^2}{25}$$
@robjohn because of the beautiful symmetry we employ cot(z) identity and we're done. (and differentiate both sides)
Surely, or we simply use digamma function ... (its series form)
halirurtan, while I undestand what you mean I believe it's just formatted that way. There was no reference to a programming language in the question or instructions.
I just thought I was missing something because the variables threw me off guard -- I am used to programming and my first thought was that x=x+y kinda threw away x=-10 and reset the variable... :( lol
