Unshared zeros of Dirichlet $L$-functions

Abstract

We prove that no Dirichlet $L$-function (and more generally, no nontrivial finite linear combination of Dirichlet $L$-functions) can vanish at every zero of a fixed $L(s,\ddot{I}{‡}_0)$. At the heart of the proof is a short-window asymptotic for the twisted discrete moment ${\sum }_{\ddot{I}}x^{\ddot{I}}L(\ddot{I},\ddot{I}{‡}_1)$, where $\ddot{I}{‡}_0\ne \ddot{I}{‡}_1$ are primitive Dirichlet characters, $\ddot{I}={\hat{I}}^2+i{\hat{I}}^3$ runs over zeros of $L(s,\ddot{I}{‡}_0)$ with $T-\hat{I}”<{\hat{I}}^3\le T$, and $x\in \mathbb{Z}$. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\hat{I}”\in [T{e}^{-C\sqrt{\log T}},T/\log T]$, thus reaching windows shorter than $T(\log T{)}^{-A}$ for any fixed $A$. Notably, the main term $\frac{\ddot{I}{‡}_1(x)}{2\ddot{I}€}\hat{I}”\log T$ depends on $x$ only through the single value $\ddot{I}{‡}_1(x)$. Since distinct primitive characters are distinguished by their values, varying $x$ isolates the contribution of each $L$-function within a linear combination, and we deduce that for a positive density of $x\in \mathbb{N}$, every nontrivial combination is nonzero at some zero of $L(s,\ddot{I}{‡}_0)$ in any sufficiently high short window. The proof combines contour integration of $-\frac{L^{\prime }}{L}(1-s,\overline{\ddot{I}}{‡}_0)L(s,\ddot{I}{‡}_1)$ with short-interval estimates for the Dirichlet convolution $(\ddot{I}{‡}_0\hat{I}›)\ast \ddot{I}{‡}_1$, which derive from the classical de la Vallée Poussin zero-free region.

18 pages, 1 figure