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Vol. 52 No. 1 (2026)
Serdica Mathematical Journal

Articles

On the location of the complex conjugate zeros of the partial theta function

https://doi.org/10.55630/serdica.2026.61-84

Published 13-04-2026

  • Vladimir Petrov Kostov

Vladimir Petrov Kostov
Université Côte d'Azur

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 \(\{{\rm Re}(x)\geq 0,~1<|x|<5\}\), 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.

2020 Mathematics Subject Classification:

26A06

Keywords

partial theta function, Jacobi theta function, Jacobi triple product

Full text

Author Details

Vladimir Petrov Kostov

Université Côte d’Azur
Centre National de la Recherche Scientifique
Laboratoire de Mathematiques “Jean-Alexandre Dieudonne”
06108 Nice, France
e-mail: vladimir.kostov@unice.fr

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