The limiting distribution of Legendre paths

Ayesha Hussain; Youness Lamzouri

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arXiv · 2304.13025

The limiting distribution of Legendre paths

Ayesha Hussain·Youness Lamzouri

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Abstract

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.

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Ayesha Hussain, Youness Lamzouri. 2023-04-25. The limiting distribution of Legendre paths. https://arxiv.org/abs/2304.13025

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