# What's the difference between a Riemann theta and a Siegel theta function?

One of the things I'm working on has required me to look into the literature of multidimensional theta functions, and I've gotten a bit confused on a few naming details.

A look at the DLMF says that "the" multidimensional theta function is the Riemann theta function,

$$\mathop{\Theta}\left(\mathbf{z}\mid\boldsymbol{{\Omega}}\right)=\sum_{{\mathbf{n}\in{\mathbb Z}^g}}e^{{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}}\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}}$$

with $\mathbf{z}$ a complex vector and $\mathbf \Omega$ a complex symmetric matrix with symmetric positive definite imaginary part. I find that there is a nice implementation for numerically computing this function, but it is only available in Maple and Java. Since I work in Mathematica, I decided to see if there was already something in Mathematica before going through the trouble of translating code.

Peering at the docs for Mathematica netted the Siegel theta function. However, looking at how the Siegel theta function was defined, this and the Riemann theta function seem to be the same thing!

Not having Maple to check if the results from their implementation of Riemann theta and the results from Mathematica's implementation of Siegel theta agree, and not being able to access the (older) references to these functions, I now wish to ask: is there no difference whatsoever between these two except in name (and if there are, how are these two different)? Did Riemann and Siegel independently study the exact same multidimensional theta function? How come it seems that there is no reference to these two being synonymous?

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Two notes: 1. attempts to search with Google had my results contaminated with stuff related to the Riemann-Siegel functions associated with the zeta function. 2. I asked here instead of on m.SE, because I thought it was more likely I'd see somebody here with theta function expertise. Hopefully this isn't too basic! –  J. M. May 8 '11 at 6:57

It's the same function. I checked with Maple 15:

$r := RiemannTheta([0.5+I, 2 I], Matrix(2, 2, [[I, 0], [0, I]]), [])$

$evalf(r)$

$-6.586149971*10^6-2.132900065*10^{-8} I$

Mathematica 8 gives

$N[SiegelTheta[{{I, 0}, {0, I}}, {0.5 + I, 2 I}]]$

$-6.58615*10^6 - 8.06571*10^{-10} I$

as Deconinck et al. write: "There are as many different conventions for writing the Riemann theta function as there are names for it."

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Interesting, thanks! (I'm fresh out of votes at the moment, but I'll upvote this later.) I guess that leaves my (historical) question of "did Riemann and Siegel independently come up with the exact same beastie?" –  J. M. May 8 '11 at 21:11
Riemann came first (1857), and Siegel generalized (1935). Some of the history, with references to the original literature, can be found here: kryakin.com/files/Invent_mat_(2_8)/31/1231_12.pdf –  Carlo Beenakker May 9 '11 at 9:02
Thanks a lot, Carlo! –  J. M. May 9 '11 at 11:11