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.
Sokal, The leading root of the partial theta function, Adv
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
I study the leading root x_0(y) of the partial theta function \Theta_0(x,y) = \sum_{n=0}^\infty x^n y^{n(n-1)/2}, considered as a formal power series. I prove that all the coefficients of -x_0(y) are strictly positive. Indeed, I prove the stronger results that all the coefficients of -1/x_0(y) after the constant term 1 are strictly negative, and all the coefficients of 1/x_0(y)^2 after the constant term 1 are strictly negative except for the vanishing coefficient of y^3.
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math.CA 2verdicts
UNVERDICTED 2representative citing papers
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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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.