On the location of the complex conjugate zeros of the partial theta function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917520948068352 |
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| author | Kostov, Vladimir Petrov |
| author_facet | Kostov, Vladimir Petrov |
| contents | We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ with non-negative real part belong to the half-annulus $\{$Re$(x)\geq 0,~1<|x|<5\}$, where the outer radius cannot be replaced by a number smaller than $e^{π/2}=4.810477382\ldots$, and that for $q\in (0,0.2^{1/4}=0.6687403050\ldots ]$, $θ(q,.)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_15866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the location of the complex conjugate zeros of the partial theta function Kostov, Vladimir Petrov Classical Analysis and ODEs 26A06 We prove that for any $q\in (0,1)$, all complex conjugate pairs of zeros of the partial theta function $θ(q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ with non-negative real part belong to the half-annulus $\{$Re$(x)\geq 0,~1<|x|<5\}$, where the outer radius cannot be replaced by a number smaller than $e^{π/2}=4.810477382\ldots$, and that for $q\in (0,0.2^{1/4}=0.6687403050\ldots ]$, $θ(q,.)$ 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. |
| title | On the location of the complex conjugate zeros of the partial theta function |
| topic | Classical Analysis and ODEs 26A06 |
| url | https://arxiv.org/abs/2501.15866 |