Borel lemma: geometric progression and zeta-functions
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866915298296201216 |
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| author | Han, Qi Liu, Jingbo Malik, Nadeem |
| author_facet | Han, Qi Liu, Jingbo Malik, Nadeem |
| contents | In the proof of the classical Borel lemma \cite{eB} by Hayman \cite{wkH}, each continuous increasing function $T(r)\geq1$ satisfies $T\bigl(r+\frac{1}{T(r)}\bigr)<2T(r)$ outside a possible exceptional set of linear measure $2$. We note in this work $T(r)$ satisfies a sharper inequality $T\bigl(r+\frac{1}{T(r)}\bigr)<\bigl(\sqrt{T(r)}+1\bigr)^2\leq2T(r)$, if $T(r)\geq\bigl(\sqrt{2}+1\bigr)^2$, outside a possible exceptional set of linear measure $ζ\bigl(2,\sqrt{2}+1\bigr)\leq0.52<2$ for the Hurwitz zeta-function $ζ(s,a)$. This result is worth noting, provided the set of $r$ in which $1\leq T(r)<\bigl(\sqrt{2}+1\bigr)^2$ has linear measure less than $1.48$. Focusing exclusively on meromorphic functions of infinite order, we utilize Hinkkanen's Second Main Theorem \cite{aH}, draw comparisons with Borel \cite{eB}, Nevanlinna \cite{rN}, and Hayman \cite{wkH}, and finally generalize Fernández Árias \cite{aFA1}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Borel lemma: geometric progression and zeta-functions Han, Qi Liu, Jingbo Malik, Nadeem Classical Analysis and ODEs 11M06, 11M35, 26A12, 30D35 In the proof of the classical Borel lemma \cite{eB} by Hayman \cite{wkH}, each continuous increasing function $T(r)\geq1$ satisfies $T\bigl(r+\frac{1}{T(r)}\bigr)<2T(r)$ outside a possible exceptional set of linear measure $2$. We note in this work $T(r)$ satisfies a sharper inequality $T\bigl(r+\frac{1}{T(r)}\bigr)<\bigl(\sqrt{T(r)}+1\bigr)^2\leq2T(r)$, if $T(r)\geq\bigl(\sqrt{2}+1\bigr)^2$, outside a possible exceptional set of linear measure $ζ\bigl(2,\sqrt{2}+1\bigr)\leq0.52<2$ for the Hurwitz zeta-function $ζ(s,a)$. This result is worth noting, provided the set of $r$ in which $1\leq T(r)<\bigl(\sqrt{2}+1\bigr)^2$ has linear measure less than $1.48$. Focusing exclusively on meromorphic functions of infinite order, we utilize Hinkkanen's Second Main Theorem \cite{aH}, draw comparisons with Borel \cite{eB}, Nevanlinna \cite{rN}, and Hayman \cite{wkH}, and finally generalize Fernández Árias \cite{aFA1}. |
| title | Borel lemma: geometric progression and zeta-functions |
| topic | Classical Analysis and ODEs 11M06, 11M35, 26A12, 30D35 |
| url | https://arxiv.org/abs/2401.14481 |