Tagged Questions
41
votes
3answers
592 views
Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$
I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$.
I found a solution by myself 10 hours after I posted it, here it is:
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0
votes
0answers
30 views
References for the conformal equivalence of the space of complex 1-tori and C?
What are some good references with proofs of the conformal equivalence of the space of complex tori and $\mathbb{C}$? So far I only have the book by Jones and Singerman.
5
votes
1answer
78 views
Direct proof of the non-zeroness of an Eisenstein series
Question: Can you show directly from its formula that $G_4(i)\neq0$?
Recall that the holomorphic Eisenstein series of weight $2k$ is defined by:
$$G_{2k}(\tau)= \sum_{(m,n)\in\mathbb{Z}^2\setminus ...
4
votes
2answers
115 views
Klein's j-invariant and Ford circles
Klein's j-invariant has structure which seems to resemble Ford circles:
The latter show up all over number theory (continued fractions, Rademacher's expansion for p(n), etc.)
Can someone explain ...
14
votes
1answer
238 views
The importance of modular forms
I'm studying modular forms and my professor started the course talking about elliptic functions. These particular functions form a field (once that the lattice $\Lambda$ is fixed) called ...
