Tagged Questions
0
votes
0answers
43 views
Interpretation of functional equation of dedekind eta function
It's well known that the Dedekind eta function defined by $\eta(z) = \displaystyle e^{\frac{\pi i z}{12}} \prod_{n=1}^{\infty} (1 - e^{2 \pi i n z}) $ converges for $z$ in the upper half plane to a ...
9
votes
0answers
160 views
Convexity of $\theta(q)$
Define Jacobi's (fourth) theta function with argument zero and nome $q$:
$$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$
plot of the function via Wolfram|Alpha
plot of the function via Sage
I ...
6
votes
1answer
528 views
On the growth of the Jacobi theta function
So, I ran into this exercise from Stein & Shakarchi. CA, Chapter 5:
Show that if $\tau$ is fixed with positive imaginary part, then the Jacobi theta function
$$\theta(z | r) = ...
1
vote
0answers
130 views
What is meant by a set of generic points on a compact Riemann surface
Let $X$ be a compact connected Riemann surface of genus $g \geq 1$.
I'm studying a theorem of Faltings which looks as follows.
Let $P_1,\ldots, P_g$ be generic points on $X$. Then we have some ...
5
votes
1answer
322 views
Where are this kind of series used, $\vartheta_{4}(0,e^{\alpha \cdot z})$?
In my recent explorations I stumbled upon the following series
$$
\vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z ...
