Strong Invariance Principle for Random Dirichlet Series
Abstract
It is shown that the random prime Dirichlet series $ X(s)=\sum_{p\in \mathcal{P}}\frac{ξ_p}{p^{s+1/2}} $ can be almost surely approximated by a Brownian motion, where $ \{ξ_p\}_{p\ge 1} $ are independent and identically distributed random variables with zero mean and finite variance, and $ \mathcal{P} $ is the set of prime numbers. As a consequence, the functional law of the iterated logarithm, limit laws of the iterated logarithm, and Chung's law of the iterated logarithm for $ X(s) $ are obtained. It is also shown that the random Dirichlet series $ Y(s)=\sum_{k=2}^{\infty} \frac{(\log k)^α}{k^{s+1/2}} $ with $ α=-1/2 $ admits the same approximation as $ X(s) $, and a direct relationship between $ X(s) $ and $ Y(s) $ with $ α=-1/2 $ is established. The strong invariance principle for $ Y(s) $ with $ α>-1/2 $ is also obtained.
Related
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- STRONG INVARIANCE PRINCIPLES IN DETERMINISTIC SYSTEMS
BibTeX
@misc{zhang2026strong,
title = {{Strong Invariance Principle for Random Dirichlet Series}},
author = {Li-Xin Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18353},
eprint = {2610.18353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18353},
}
RIS
TY - UNPB
TI - Strong Invariance Principle for Random Dirichlet Series
AU - Li-Xin Zhang
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.18353
UR - https://arxiv.org/abs/2610.18353
AB - It is shown that the random prime Dirichlet series $ X(s)=\sum_{p\in \mathcal{P}}\frac{ξ_p}{p^{s+1/2}} $ can be almost surely approximated by a Brownian motion, where $ \{ξ_p\}_{p\ge 1} $ are independent and identically distributed random variables with zero mean and finite variance, and $ \mathcal{P} $ is the set of prime numbers. As a consequence, the functional law of the iterated logarithm, limit laws of the iterated logarithm, and Chung's law of the iterated logarithm for $ X(s) $ are obtained. It is also shown that the random Dirichlet series $ Y(s)=\sum_{k=2}^{\infty} \frac{(\log k)^α}{k^{s+1/2}} $ with $ α=-1/2 $ admits the same approximation as $ X(s) $, and a direct relationship between $ X(s) $ and $ Y(s) $ with $ α=-1/2 $ is established. The strong invariance principle for $ Y(s) $ with $ α>-1/2 $ is also obtained.
ER -