On the positivity of some weighted partial sums of a random multiplicative function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908474178273280 |
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| author | Aymone, Marco |
| author_facet | Aymone, Marco |
| contents | Inspired by the papers by Angelo and Xu, Q.J Math., 74, pp. 767-777, and improvements by Kerr and Klurman, arXiv:2211.05540, we study the probability that the weighted sums of a Rademacher random multiplicative function, $\sum_{n\leq x}f(n)n^{-σ}$, are positive for all $x\geq x_σ\geq 1$ in the regime $σ\to1/2^+$. In a previous paper by Heap, Zhao and the author, and by the author, when $0\leq σ\leq 1/2$ this probability is zero. Here we give a positive lower bound for this probability depending on $x_σ$ that becomes large as $σ\to1/2^+$. The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Bonami--Halász's moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15589 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the positivity of some weighted partial sums of a random multiplicative function Aymone, Marco Number Theory Probability Inspired by the papers by Angelo and Xu, Q.J Math., 74, pp. 767-777, and improvements by Kerr and Klurman, arXiv:2211.05540, we study the probability that the weighted sums of a Rademacher random multiplicative function, $\sum_{n\leq x}f(n)n^{-σ}$, are positive for all $x\geq x_σ\geq 1$ in the regime $σ\to1/2^+$. In a previous paper by Heap, Zhao and the author, and by the author, when $0\leq σ\leq 1/2$ this probability is zero. Here we give a positive lower bound for this probability depending on $x_σ$ that becomes large as $σ\to1/2^+$. The main inputs in our proofs are a maximal inequality based in relatively high moments for these partial sums combined with a Bonami--Halász's moment inequality, and also explicit estimates for the partial sums of non-negative multiplicative functions. |
| title | On the positivity of some weighted partial sums of a random multiplicative function |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2408.15589 |