On a rigidity property for quadratic gauss sums

Let be a large prime and let . We prove that if is a ‐valued multiplicative function, such that the exponential sums satisfy the ‘Gauss sum‐like’ approximate dilation symmetry property uniformly over all primes , then coincides with a real character modulo at almost all . As a consequence, taking to be the Liouville function, we connect this exponential sums property to the location of real zeros of close to , for the Legendre symbol modulo . Assuming that the ‐functions of primitive Dirichlet characters modulo have a sufficiently wide zero‐free region (of ‘Littlewood type'), we also show a more general result in which may be taken arbitrarily close to 0.