Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals
In the article, we obtain that, for algebraically independent over Q{\mathbb{Q}} parameters α1,…,αr{\alpha }_{1},\ldots ,{\alpha }_{r}, there are infinitely many shifts (ζ(s+iτ,α1),…,ζ(s+iτ,αr))\left(\zeta \left(s+i\tau ,{\alpha }_{1}),\ldots ,\zeta \left(s+i\tau ,{\alpha }_{r})) of Hurwitz zeta-fun...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-08-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2025-0173 |
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| Summary: | In the article, we obtain that, for algebraically independent over Q{\mathbb{Q}} parameters α1,…,αr{\alpha }_{1},\ldots ,{\alpha }_{r}, there are infinitely many shifts (ζ(s+iτ,α1),…,ζ(s+iτ,αr))\left(\zeta \left(s+i\tau ,{\alpha }_{1}),\ldots ,\zeta \left(s+i\tau ,{\alpha }_{r})) of Hurwitz zeta-functions with τ∈[T,T+H]\tau \in \left[T,T+H], T27⁄82⩽H⩽T1⁄2{T}^{27/82}\leqslant H\leqslant {T}^{1/2}, that approximate any rr-tuple of analytic functions on the strip {s∈C:1⁄2<σ<1}\left\{s\in {\mathbb{C}}:1/2\lt \sigma \lt 1\right\}. More precisely, the latter set of shifts has a positive density. For the proof, a probabilistic approach is applied. |
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| ISSN: | 2391-5455 |