Large values of derivatives of quadratic Dirichlet $L$-functions

Source abstract

e study large values of fixed derivatives of quadratic Dirichlet $L$-functions on the real segment $1/2\leqσ\leq1$. Assuming GRH, we prove lower bounds in terms of GCD sums and finite Euler products. At the central point, every fixed derivative order is compared with the same square-free GCD sum. In the mesoscopic band $σ=1/2+A/\log_2X$, a renormalized resonator estimates contributions from both halves of the differentiated approximate functional equation. The resulting lower bound has a relative error. On a smaller boundary scale, a common finite product appears in both contributions to this lower bound. This allows us to locate the crossover between their derivative weights. The coarser exponential bounds have constant $1$ at the center and $\mathrm e^{-A}$ in the mesoscopic band. For fixed $1/2<σ<1$, the lower bound includes the full typical logarithmic weight of the resonating Euler product. Burgess truncation improves the exponential constant when a shorter cutoff allows a longer resonator. Near the $1$-line, our lower bounds involve Dickman constants on the $1/\log_2X$ scale. We also prove a uniform transition estimate outside a shrinking neighborhood of the central point. All results hold in the positive even subfamily of fundamental discriminants.