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Limit theorems for random Dirichlet series: boundary case
Alexander Iksanov ORCID icon link to view author Alexander Iksanov details   Ruslan Kostohryz  

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https://doi.org/10.15559/25-VMSTA276
Pub. online: 25 March 2025      Type: Research Article      Open accessOpen Access

Received
4 November 2024
Revised
25 January 2025
Accepted
6 March 2025
Published
25 March 2025

Abstract

Buraczewski et al. (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series ${\textstyle\sum _{k\ge 2}}\frac{{(\log k)^{\alpha }}}{{k^{1/2+s}}}{\eta _{k}}$ as $s\to 0+$, where $\alpha \gt -1/2$ and ${\eta _{1}},{\eta _{2}},\dots $ are independent identically distributed random variables with zero mean and finite variance. A FLT and a LIL are proved in a boundary case $\alpha =-1/2$. The boundary case is more demanding technically than the case $\alpha \gt -1/2$. A FLT and a LIL for ${\textstyle\sum _{p}}\frac{{\eta _{p}}}{{p^{1/2+s}}}$ as $s\to 0+$, where the sum is taken over the prime numbers, are stated as the conjectures.

References

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© 2025 The Author(s). Published by VTeX
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Open access article under the CC BY license.

Keywords
Functional limit theorem law of the iterated logarithm random Dirichlet series

MSC2020
60F15 60F17 60G50

Funding
The present work was supported by the National Research Foundation of Ukraine (project 2023.03/0059 ‘Contribution to modern theory of random series’).

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