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.
St. Kliment Ohridski
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All complex conjugate zeros of θ(q,x) with Re(x)≥0 lie in 1<|x|<5 for q∈(0,1), none exist for q≤0.6687..., and those with Re(x)<0 lie in |x|<49.8.
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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On the location of the complex conjugate zeros of the partial theta function
All complex conjugate zeros of θ(q,x) with Re(x)≥0 lie in 1<|x|<5 for q∈(0,1), none exist for q≤0.6687..., and those with Re(x)<0 lie in |x|<49.8.
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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|.