# Why do the Jacobi theta functions have a natural boundary?

The Jacobi theta functions, like $$\theta_3(z,q)=1+2\sum_{n=0}^\infty q^{n^2}\!\cos(2nz) ,$$ look relatively innocent in how they handle the 'nome' $q$, a complex parameter that shapes the dependence of $\theta_3$ on its argument $z$. At a first look, the series is definitely convergent if $|q|<1$, with some vague hopes of not-horrible behaviour at the edge of the unit disk. However, as it turns out, not only does the series in the definition above diverge for $|q|>1$, but the theta function simply cannot be analytically continued past the edge of the unit disk, where it has a natural boundary.

I would like to understand exactly how the existence of this natural boundary, and the impossibility of analytical continuation beyond it, arises and how it can be rigorously proved. What is the cleanest, most flexible way of showing it's there?

(In particular, I would like to decide whether a similar series, $\sum_{n=-\infty}^\infty\exp(i(an^3+bn^2+cn))$, has a similar behaviour with respect to the equivalent parameter $q=e^{ib}$, so I would appreciate results which can be extended in that sort of direction.)

• A bit of terminology: A function which can't be analytically continued outside the circle of converence is known as a lacunary function. This article (pdf form) discusses them in detail, and includes some mention of the Jacobi case: people.math.osu.edu/costin.9/f61lacunary.pdf. – Semiclassical Jun 16 '16 at 19:45
• Also, the Ostrowski-Hadamard gap theorem is quite pertinent here. (I don't know how it's proven, but it should be available online somewhere...) This blog post also mentions a version by Fabry which is sufficient for the Jacobi theta function: uniformlyatrandom.wordpress.com/2009/03/28/… – Semiclassical Jun 16 '16 at 19:57
• @Semiclassical The Fabry gap theorem looks strong enough - I'm happy to accept a good reference as an answer. (The Ostrowski-Hadamard gap theorem, on the other hand, is not quite strong enough for the Jacobi case.) – E.P. Jun 16 '16 at 20:09