On a rigidity property for quadratic Gauss sums

2025/02/22 by Mangerel, Alexander P.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.16014

Abstract

Let N be a large prime and let c > 1/4. We prove that if f is a ± 1-valued completely multiplicative function, such that the exponential sums S_f(a) := ∑_{1 ≤ n lt; N} f(n) e(na/N), a \pmodN satisfy the ``Gauss sum-like'' approximate dilation symmetry property (1)/(N)∑_a \pmodN |S_f(ap) - f(p)S_f(a)|² = o(N), uniformly over all primes p ≤ N^c then f coincides with a real character modulo N at all but o(N) integers 1 ≤ n < N. As a consequence, taking f to be the Liouville function we connect this exponential sums property to the location of real zeros of L(s,χ) close to s = 1, for χ the Legendre symbol modulo N. Assuming the L-functions of primitive Dirichlet characters modulo N have a sufficiently wide zero-free region (of Littlewood type), we also show a more general result in which any c > 0 may be taken.

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