Non-vanishing of high derivatives of Dirichlet \(L\)-functions at the central point
Let \(\chi\) run over all even Dirichlet characters to modulus \(q\), and let \(\Lambda(s,\chi)\) be the `completed' \(L\)-function. It is shown that if \(q\) is large and \(k\) is a fixed positive integer, then a proportion at least \(c_k\) of the derivatives \(\Lambda^{(k)} (\frac 12,\chi)\) are nonzero, with constants \(c_k= \frac 23- O(k^{-2})\). The authors say that a similar result can be proved for odd characters. One can think of this as being analogous to \textit{B. Conrey}'s result [J. Number Theory 16, 49-74 (1983; Zbl 0502.10022)] for the Riemann \(\xi\)-function, which states that a proportion \(1- O(k^{-2})\) of the zeros of \(\xi^{(k)}(s)\) lie on the critical line. It is however disappointing that the constants \(c_k\) only tend to \(\frac 23\) in the present case. The proof depends on a comparison of the first and second moments of \(\Lambda^{(k)} (\frac 12,\chi) M_k (\frac 12,\chi)\), where \(M(s,\chi)\) is a mollifier, taken to have the form \[ M^*(s,\chi)+ (-1)^k \frac{\overline{\tau(\chi)}} {\sqrt{q}} M^*(1-s, \overline{\chi}) \] for a suitable Dirichlet polynomial \(M^*(s,\chi)\).
scientific article; zbMATH DE number 1420268