Non-vanishing of \(n\)-th derivatives of twisted elliptic \(L\)-functions in the critical point (Q1382088)

The author shows that \[ N_n(D)\gg D^{1-\varepsilon}, \] where \(\varepsilon >0\), \(N_n(D)\) is the number of \(d\leq D\) (satisfying some arithmetical conditions) such that \(L^{(n)} (1,E_d) \neq 0\), \(E_d\) being the twist of a modular elliptic curve \(E\) over \(\mathbb{Q}\), and \(L(s,E_d)\) its associated zeta function. This is the extension to the \(n\)-th derivative of a recent result by \textit{J. Pomykała} and \textit{A. Perelli} [Acta Arith. 80, No. 2, 149-163; (1997; Zbl 0878.11022)] in the case \(n=1\). The general case is based on the same ingredients as the case \(n=1\), i.e., an \(L'\)-norm evaluation and an \(L^2\)-norm estimate related with \(L(1,E_d)\). The author follows \textit{H. Iwaniec}'s [Sémin. Théor. Nombres Bordx., Sér. II 2, No. 2, 365-376 (1990; Zbl 0719.11029)] approach for the \(L'\)-norm evaluation, while the \(L^2\)-norm estimate is obtained by the technique in the author-reviewer paper above. (The title of this paper is misquoted in the references).