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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De Gruyter
2025-08-01
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| Series: | Open Mathematics |
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| Online Access: | https://doi.org/10.1515/math-2025-0173 |
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| author | Laurinčikas Antanas Šiaučiūnas Darius |
| author_facet | Laurinčikas Antanas Šiaučiūnas Darius |
| author_sort | Laurinčikas Antanas |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-60cb5e1c113f4b6a9bee7a83bb3fbe29 |
| institution | Kabale University |
| issn | 2391-5455 |
| language | English |
| publishDate | 2025-08-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Open Mathematics |
| spelling | doaj-art-60cb5e1c113f4b6a9bee7a83bb3fbe292025-08-20T03:44:06ZengDe GruyterOpen Mathematics2391-54552025-08-012318610110.1515/math-2025-0173Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervalsLaurinčikas Antanas0Šiaučiūnas Darius1Faculty of Mathematics and Informatics, Institute of Mathematics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, LithuaniaInstitute of Regional Development, Šiauliai Academy, Vilnius University, Vytauto str. 84, LT-76352 Šiauliai, LithuaniaIn 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.https://doi.org/10.1515/math-2025-0173hurwitz zeta-functionjoint universalityspace of analytic functionsweak convergence of probability measures11m35 |
| spellingShingle | Laurinčikas Antanas Šiaučiūnas Darius Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals Open Mathematics hurwitz zeta-function joint universality space of analytic functions weak convergence of probability measures 11m35 |
| title | Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals |
| title_full | Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals |
| title_fullStr | Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals |
| title_full_unstemmed | Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals |
| title_short | Joint approximation of analytic functions by the shifts of Hurwitz zeta-functions in short intervals |
| title_sort | joint approximation of analytic functions by the shifts of hurwitz zeta functions in short intervals |
| topic | hurwitz zeta-function joint universality space of analytic functions weak convergence of probability measures 11m35 |
| url | https://doi.org/10.1515/math-2025-0173 |
| work_keys_str_mv | AT laurincikasantanas jointapproximationofanalyticfunctionsbytheshiftsofhurwitzzetafunctionsinshortintervals AT siauciunasdarius jointapproximationofanalyticfunctionsbytheshiftsofhurwitzzetafunctionsinshortintervals |