Sign changes of short character sums and real zeros of Fekete polynomials

Abstract

We discuss a general approach producing quantitative bounds on the number of sign changes of the weighted sums $$\sum _{n\le x}f(n)w_n$$ where $f:\mathbb{N}\to \mathbb{R}$ is a family of multiplicative functions and $w_n\in \mathbb{R}$ are certain weights.As a consequence, we show that for a typical fundamental discriminant $D,$ the partial sums of the real character ${\chi }_D$ change sign $\gg (\log \log D)/\log \log \log \log D$ times on very short initial interval (which goes beyond the range in Vinogradov's conjecture). We also prove that the number of real zeros (localized away from $1$) of the Fekete polynomial associated to a typical fundamental discriminant $D$ is $\gg \frac{\log \log D}{\log \log \log \log D}.$ This comes close to establishing a conjecture of Baker and Montgomery which predicts $\asymp \log \log D$ real zeros. Finally, the same approach shows that almost surely for large $x\ge 1$, the partial sums ${\sum }_{n\le y}f(n)$ of a (Rademacher) random multiplicative function exhibit $\gg \log \log x/\log \log \log \log x$ sign changes on the interval $[1,x].$ These results rely crucially on uniform quantitative estimates for the joint distribution of $-\frac{L^{\prime }}{L}(s,{\chi }_D)$ at several points $s$ in the vicinity of the central point $s=1/2$, as well as concentration results for $\log L(s,{\chi }_D)$ in the same range, which we establish. In the second part of the paper, we obtain, for large families of discriminants, ``non-trivial" upper bounds on the number of real zeros of Fekete polynomials, breaking the square root bound. Finally, we construct families of discriminants with associated Fekete polynomials having no zeros away from $1.$