The limiting distribution of Legendre paths

Let p be a prime number and (· p) be the Legendre symbol modulo p. The Legendre path attached to p is the polygonal path whose vertices are the normalized character sums 1 p∑ n≤j (n p) for 0≤j≤p-1. In this paper, we investigate the distribution of Legendre paths as we vary over the primes p such that Q≤p≤2Q, when Q is large. Our main result shows that as Q→∞, 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.