arXiv subjects
Publications and source records attributed to Xinghua Cui.
In this paper, we establish lower bounds for extreme values of derivatives of Dirichlet \(L\)-functions in the range \(1/2<\sigma<1\). Compared with the work of Aistleitner, Mahatab, Munsch, and Peyrot in 2019, our result shows that derivatives of Dirichlet \(L\)-functions can attain extreme values of the same order of magnitude as the original \(L\)-functions when \(1/2<\sigma<1\) holds.
The trace formula constitutes a fundamental tool in the Langlands program. In general, Arthur introduced a truncation operator to render both the geometric and spectral sides of the formula convergent. This paper focuses on the case of $\mathrm{GL}(3)$. We first prove that the divergent terms on the geometric and spectral sides are equal, leading to their cancellation. We derive an explicit formula for ramified orbital integrals, showing they are limits of unramified ones and that Arthur's definition yields a universal object, agreeing with that of Hoffmann-Wakatsuki. Finally, on the spectral side, we apply normalized intertwining operators to present the expansion in a form parallel to that of the geometric side.
Trace formula is an important method to study the Langlands program. Arthur obtains the existence of stable trace formula for connected reductive group. In this paper, we will give the explicit coarse trace formula of GL(4). In general case, Arthur applies the truncation operator on the two sides of trace formula, which is convergent. In our case, we will prove that the divergent terms of the two sides of the trace formula of GL(4)$ are equal. We also obtain the explicit formula for ramified orbits of the geometric side of trace formula of GL(4).