Two properties of the partial theta function
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866918250652106752 |
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| author | Kostov, Vladimir Petrov |
| author_facet | Kostov, Vladimir Petrov |
| contents | For the partial theta function $θ(q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$, $q$, $z\in \mathbb{C}$, $|q|<1$, we prove that its zero set is connected. This set is smooth at every point $(q^{\flat},z^{\flat})$ such that $z^{\flat}$ is a simple or double zero of $θ(q^{\flat},.)$. For $q\in (0,1)$, $q\rightarrow 1^-$ and $a\geq e^π$, there are $o(1/(1-q))$ and $(\ln (a/e^π))/(1-q)+o(1/(1-q))$ real zeros of $θ(q,.)$ in the intervals $[-e^π,0)$ and $[-a,-e^{-π}]$ respectively (and none in $[0,\infty)$). For $q\in (-1,0)$, $q\rightarrow -1^+$ and $a\geq e^{π/2}$, there are $o(1/(1+q))$ real zeros of $θ(q,.)$ in the interval $[-e^{π/2},e^{π/2}]$ and $(\ln (a/e^{π/2})/2)/(1+q)+o(1/(1+q))$ in each of the intervals $[-a,-e^{π/2}]$ and $[e^{π/2},a]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1911_08841 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Two properties of the partial theta function Kostov, Vladimir Petrov Classical Analysis and ODEs For the partial theta function $θ(q,z):=\sum_{j=0}^{\infty}q^{j(j+1)/2}z^j$, $q$, $z\in \mathbb{C}$, $|q|<1$, we prove that its zero set is connected. This set is smooth at every point $(q^{\flat},z^{\flat})$ such that $z^{\flat}$ is a simple or double zero of $θ(q^{\flat},.)$. For $q\in (0,1)$, $q\rightarrow 1^-$ and $a\geq e^π$, there are $o(1/(1-q))$ and $(\ln (a/e^π))/(1-q)+o(1/(1-q))$ real zeros of $θ(q,.)$ in the intervals $[-e^π,0)$ and $[-a,-e^{-π}]$ respectively (and none in $[0,\infty)$). For $q\in (-1,0)$, $q\rightarrow -1^+$ and $a\geq e^{π/2}$, there are $o(1/(1+q))$ real zeros of $θ(q,.)$ in the interval $[-e^{π/2},e^{π/2}]$ and $(\ln (a/e^{π/2})/2)/(1+q)+o(1/(1+q))$ in each of the intervals $[-a,-e^{π/2}]$ and $[e^{π/2},a]$. |
| title | Two properties of the partial theta function |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/1911.08841 |