For the partial theta function θ(q,x), real zeros lie left of a vertical line Re x = -a (a≥5) while complex zeros lie right of it, with no real zeros ≥-6 for q>0 and similar bounds for q<0.
Kostov, On the location of the complex conjugate zeros of the partial theta function, Serdica Math
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abstract
We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ with non-negative real part belong to the half-annulus $\{$Re$(x)\geq 0,~1<|x|<5\}$, where the outer radius cannot be replaced by a number smaller than $e^{\pi /2}=4.810477382\ldots$, and that for $q\in (0,0.2^{1/4}=0.6687403050\ldots ]$, $\theta (q,.)$ has no zeros with non-negative real part. The complex conjugate pairs of zeros with negative real part belong to the left open half-disk of radius $49.8$ centered at the origin.
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math.CA 2years
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Absence of spectral values (q with multiple zeros of partial theta) proven in sector union disk radius 0.207875..., with one value at 0.309249... and zero-moduli separation by negative half-integer powers of |q|.
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Some analytic properties of the partial theta function
For the partial theta function θ(q,x), real zeros lie left of a vertical line Re x = -a (a≥5) while complex zeros lie right of it, with no real zeros ≥-6 for q>0 and similar bounds for q<0.
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Where not to find the spectrum of the partial theta function
Absence of spectral values (q with multiple zeros of partial theta) proven in sector union disk radius 0.207875..., with one value at 0.309249... and zero-moduli separation by negative half-integer powers of |q|.