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I was reading Ahlfors' and came across the "genus of the canonical product." I'm curious if there's any relationship between this genus and the genus of a torus(if so, my best guess is that these form a level set on a torus of the same genus). Or, is this just a case of, "we needed a name, and everything's been taken?"


Just to be clear, what I'm referencing is pg 196:

"The preceding proof has shown that the product

$$ (22) \hspace{0.5cm}\prod_{1}^{\infty}\left(1-\frac{z}{a_{n}}\right) e^{\frac{z}{a_{n}}+\frac{1}{2}\left(\frac{z}{a_{n}}\right)^{2}+\cdots+\frac{1}{h}\left(\frac{z}{a_{n}}\right)^{h}} $$ converges and represents an entire function provided that the series $\sum_{n=1}^{\infty}\left(R /\left|a_{n}\right|\right)^{h+1} /(h+1)$ converges for all $R,$ that is to say provided that $\Sigma 1 /\left|a_{n}\right|^{h+1}<\infty .$ Assume that $h$ is the smallest integer for which this series converges; the expression (22) is then called the canonical product associated with the sequence $\left\{a_{n}\right\},$ and $h$ is the genus of the canonical product."

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  • $\begingroup$ Thoughts, anyone? $\endgroup$ Mar 10, 2021 at 0:57
  • $\begingroup$ I do not think this notion is related in any way to topology. It happens frequently that X in one area of math has no relation to X in another. For instance, fields in Algebra have nothing to do with vector fields, normality in algebra is unrelated to normality in topology. I remember increasingly weird conversation with another mathematician discussing $H^2$: After 5 minutes, it turned out that I was talking about hyperbolic plane and he meant a Sobolev space (it could have been the 2nd cohomology group). $\endgroup$ Mar 10, 2021 at 3:50
  • $\begingroup$ That's kind of what I was assuming but had to check. $\endgroup$ Mar 11, 2021 at 20:17

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