The limiting distribution of Legendre paths
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866916234724900864 |
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| author | Hussain, Ayesha Lamzouri, Youness |
| author_facet | Hussain, Ayesha Lamzouri, Youness |
| contents | Let $p$ be a prime number and $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol modulo $p$. The \emph{Legendre path} attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}} \sum_{n\leq j} \left(\frac{n}{p}\right)$ for $0\leq j\leq p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $Q\leq p\leq 2Q$, when $Q$ is large. Our main result shows that as $Q \to \infty$, these paths converge in law, in the space of real-valued continuous functions on $[0, 1]$, to a certain random Fourier series constructed using Rademacher random completely multiplicative functions. This was previously proved by the first author under the assumption of the Generalized Riemann Hypothesis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_13025 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The limiting distribution of Legendre paths Hussain, Ayesha Lamzouri, Youness Number Theory Probability Let $p$ be a prime number and $\left(\frac{\cdot}{p}\right)$ be the Legendre symbol modulo $p$. The \emph{Legendre path} attached to $p$ is the polygonal path whose vertices are the normalized character sums $\frac{1}{\sqrt{p}} \sum_{n\leq j} \left(\frac{n}{p}\right)$ for $0\leq j\leq p-1$. In this paper, we investigate the distribution of Legendre paths as we vary over the primes $Q\leq p\leq 2Q$, when $Q$ is large. Our main result shows that as $Q \to \infty$, these paths converge in law, in the space of real-valued continuous functions on $[0, 1]$, to a certain random Fourier series constructed using Rademacher random completely multiplicative functions. This was previously proved by the first author under the assumption of the Generalized Riemann Hypothesis. |
| title | The limiting distribution of Legendre paths |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2304.13025 |