Unshared zeros of Dirichlet $L$-functions
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 $L(s,\chi_0)$. At the heart of the proof is a short-window asymptotic for the twisted discrete moment $\sum_{\rho} x^{\rho} L(\rho,\chi_1)$, where $\chi_0\ne\chi_1$ are primitive Dirichlet characters, $\rho=\beta+i\gamma$ runs over zeros of $L(s,\chi_0)$ with $T-\Delta<\gamma\le T$, and $x\in\mathbb{Z}$. The asymptotic is unconditional, assuming no hypothesis of GRH type, and it holds for every window width $\Delta\in[T e^{-C\sqrt{\log T}},\,T/\log T]$, thus reaching windows shorter than $T(\log T)^{-A}$ for any fixed $A$. Notably, the main term $\frac{\chi_1(x)}{2\pi}\,\Delta\log T$ depends on $x$ only through the single value $\chi_1(x)$. 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 $L(s,\chi_0)$ in any sufficiently high short window. The proof combines contour integration of $-\frac{L'}{L}(1-s,\overline{\chi}_0)\,L(s,\chi_1)$ with short-interval estimates for the Dirichlet convolution $(\chi_0\Lambda)*\chi_1$, which derive from the classical de la Vall\'ee Poussin zero-free region.