The spectrum of the partial theta function accumulates at every point on the unit circle and is locally finite inside |q| ≤ 0.8, with a truncation-seeded Newton method and radial monodromy for computation.
Where not to find the spectrum of the partial theta function
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abstract
The spectrum of Ramanujan's partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which $\theta (q,.)$ has a multiple zero. We show that there is no spectral value in the set $\mathbb{S}\cup \mathbb{D}_{c_0}$, $c_0=0.2078750206\ldots$, where $\mathbb{S}$ is the sector $\{ 0<|z|<0.6,{\rm arg}(z)\in [\pi /4 ,7\pi /4 ]\}$. There is a single spectral value in the set $\mathbb{S}\cup \mathbb{D}_{0.31}$ which equals $0.309249\ldots$. For $q\in \mathbb{S}\cup \mathbb{D}_{c_0}$, the moduli of the zeros of $\theta$ are separated by the negative half-integer powers of $|q|$.
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2026 1verdicts
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Complex spectrum of the partial theta function
The spectrum of the partial theta function accumulates at every point on the unit circle and is locally finite inside |q| ≤ 0.8, with a truncation-seeded Newton method and radial monodromy for computation.