Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions

In this paper, we establish a new lower bound for the number of low-lying zeros of Dirichlet $L$-functions $L(s, \chi)$ on the critical line within extremely short intervals. Specifically, for a sufficiently large prime $P$ and real number $T \in [a_1/\sqrt{\log P}, 1]$, we prove that the sum of the number of zeros on the critical line $N_0(T, \chi)$ over characters $\chi \bmod P$ satisfies $$\sum_{\chi \bmod P} N_0(T, \chi) \gg T^2 P\sqrt{\log P} .$$ Traditional approaches encounter significant technical barriers in this short-interval regime. The Levinson method fails due to its own inherent limitations in handling such restricted intervals , while standard applications of the Selberg mollifier are hindered by the emergence of complex, inseparable cross-terms that are difficult to evaluate. To overcome these obstacles, we introduce a novel analytic framework utilizing high-dimensional Mellin transforms. This approach systematically manages the multi-variable series generated by the mollifier calculations. By explicitly resolving these cross-term obstructions, we extract the localized lower bound, providing a robust method that circumvents the short-interval bottleneck and offers potential applicability to the zero statistics of higher-rank $L$-function families.