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.
Partial theta functions and mock modular forms as q-hypergeometric series
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abstract
Ramanujan studied the analytic properties of many $q$-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious $q$-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have $q$-expansions resembling modular theta functions, is not well understood. Here we consider families of $q$-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.
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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.