Where not to find the spectrum of the partial theta function
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918529487339520 |
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| author | Gati, Yousra Kostov, Vladimir Petrov |
| author_facet | Gati, Yousra Kostov, Vladimir Petrov |
| contents | The spectrum of Ramanujan's partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which $θ(q,.)$ has a multiple zero. We show that there is no spectral value in the set $\mathbb{S}\cup \mathbb{D}_{c_0}$, $c_0=0.2078750206\ldots$, where $\mathbb{S}$ is the sector $\{ 0<|z|<0.6,{\rm arg}(z)\in [π/4 ,7π/4 ]\}$. There is a single spectral value in the set $\mathbb{S}\cup \mathbb{D}_{0.31}$ which equals $0.309249\ldots$. For $q\in \mathbb{S}\cup \mathbb{D}_{c_0}$, the moduli of the zeros of $θ$ are separated by the negative half-integer powers of $|q|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_29903 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Where not to find the spectrum of the partial theta function Gati, Yousra Kostov, Vladimir Petrov Classical Analysis and ODEs The spectrum of Ramanujan's partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$, $q\in \mathbb{D}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which $θ(q,.)$ has a multiple zero. We show that there is no spectral value in the set $\mathbb{S}\cup \mathbb{D}_{c_0}$, $c_0=0.2078750206\ldots$, where $\mathbb{S}$ is the sector $\{ 0<|z|<0.6,{\rm arg}(z)\in [π/4 ,7π/4 ]\}$. There is a single spectral value in the set $\mathbb{S}\cup \mathbb{D}_{0.31}$ which equals $0.309249\ldots$. For $q\in \mathbb{S}\cup \mathbb{D}_{c_0}$, the moduli of the zeros of $θ$ are separated by the negative half-integer powers of $|q|$. |
| title | Where not to find the spectrum of the partial theta function |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2605.29903 |