Short sums of the polynomial Mobius function

A classical result of Stickelberger relates the polynomial Mobius function modulo a prime $p$ to quadratic characters of polynomial discriminants. We explain how to use a recent stratification result of J. Xu (2020) to estimate the corresponding character sums over polynomials of degree $n$ with coefficients in a cubic box of side length $H$. We obtain a nontrivial bound with power saving below the Pólya–Vinogradov range, namely for $H \ge p^{1/2-\gamma_n}$, where $\gamma_n>0$ is an explicit constant. The main novelty of our approach is a “custom-made” modification of the Burgess shift, which allows us to handle multivariate character sums involving non-homogeneous polynomials (in the homogeneous case, the classical Burgess shift applies).

Motivated by work of S. Ganguly and C. S. Rajan (2023), we also apply a similar idea to investigate $2\times 2$ integral matrices with entries in $[1,H]$ and an irreducible characteristic polynomial. Our results are nontrivial for $H \ge p^{1/8+\varepsilon}$.

This is joint work with E. Fouvry and P. Xi