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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2510.25691 |
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| _version_ | 1866909968179920896 |
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| author | Kucheriaviy, Petr |
| author_facet | Kucheriaviy, Petr |
| contents | Let $f(n)$ be a random completely multiplicative function such that $f(p) = \pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p) = 1$ for all $p < y$, that all partial sums of $f(n)$ up to $x$ are nonnegative. We prove that for $y \ge C \frac{(\log x)^2 \log_2 x}{\log_3 x}$ this probability equals $1 - o(1)$. We also study the probability $P_x'$ that $\sum_{n \le x} \frac{f(n)}{n}$ is negative. We prove that $P_x' \ll \exp \left( - \exp \left( \frac{\log x \log_4 x}{(1 + o(1)) \log_3 x} \right) \right)$, which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Halász's theorem, we prove that $P_x' \ll \exp(-x^α)$ for some $α> 0$. Let $χ_p(n) = \left( \frac{n}{p} \right)$ be the Legendre symbol modulo $p$. For a prime $p$ chosen uniformly at random from $(x, 2x]$, we express the probability that all partial sums of $\frac{χ_p(n)}{n}$ are nonnegative in terms of the probability that partial sums of $\frac{f(n)}{n}$ are nonnegative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_25691 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positivity of partial sums of a random multiplicative function and corresponding problems for the Legendre symbol Kucheriaviy, Petr Number Theory Let $f(n)$ be a random completely multiplicative function such that $f(p) = \pm 1$ with probabilities $1/2$ independently at each prime. We study the conditional probability, given that $f(p) = 1$ for all $p < y$, that all partial sums of $f(n)$ up to $x$ are nonnegative. We prove that for $y \ge C \frac{(\log x)^2 \log_2 x}{\log_3 x}$ this probability equals $1 - o(1)$. We also study the probability $P_x'$ that $\sum_{n \le x} \frac{f(n)}{n}$ is negative. We prove that $P_x' \ll \exp \left( - \exp \left( \frac{\log x \log_4 x}{(1 + o(1)) \log_3 x} \right) \right)$, which improves a bound given by Kerr and Klurman. Under a conjecture closely related to Halász's theorem, we prove that $P_x' \ll \exp(-x^α)$ for some $α> 0$. Let $χ_p(n) = \left( \frac{n}{p} \right)$ be the Legendre symbol modulo $p$. For a prime $p$ chosen uniformly at random from $(x, 2x]$, we express the probability that all partial sums of $\frac{χ_p(n)}{n}$ are nonnegative in terms of the probability that partial sums of $\frac{f(n)}{n}$ are nonnegative. |
| title | Positivity of partial sums of a random multiplicative function and corresponding problems for the Legendre symbol |
| topic | Number Theory |
| url | https://arxiv.org/abs/2510.25691 |