Some analytic properties of the partial theta function
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913009780129792 |
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| author | Kostov, Vladimir Petrov |
| author_facet | Kostov, Vladimir Petrov |
| contents | We prove new properties of the zero set of Ramanujan's partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$, $a\geq 5$, such that all real zeros of $θ(q,.)$ lie to its left and all complex zeros to its right. A similar property is proved for $q\in (-1,0)$. For $q\in (0,1)$, there are no real zeros $\geq -6$. For $q\in (-1,0)$, there are no negative zeros $\geq -2.4$ and no positive zeros $\leq 2.4$, except the smallest one. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05559 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Some analytic properties of the partial theta function Kostov, Vladimir Petrov Classical Analysis and ODEs We prove new properties of the zero set of Ramanujan's partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in (-1,0)\cup (0,1)$, $x\in \mathbb{R}$. We show that for each $q\in (0,1)$, there exists a line Re$x=-a$, $a\geq 5$, such that all real zeros of $θ(q,.)$ lie to its left and all complex zeros to its right. A similar property is proved for $q\in (-1,0)$. For $q\in (0,1)$, there are no real zeros $\geq -6$. For $q\in (-1,0)$, there are no negative zeros $\geq -2.4$ and no positive zeros $\leq 2.4$, except the smallest one. |
| title | Some analytic properties of the partial theta function |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2604.05559 |