On the \(a\)-points of the \(k^{\text{th}}\) derivative of an \(L\)-function in the Selberg class (Q6542790)

Let \(F^{(k)}(s)\) be the \(k\)-th derivative of a function \(F(s)\) in the Selberg class. Let \(a\in \mathbb{C}\) and \(s\in \mathbb{C}\). The complex number \(s=\sigma+it\) is an \(a\)-point of \(F^{(k)}(s)\) if \(F^{(k)}(s) = a\). In this paper under review, the authors study the distribution of the \(a\)-points of \(F^{(k)}(s)\) and give an estimate for the number of these \(a\)-points. Actually, they show for sufficiently large \(T\) \N\[\NN_k(a, T, F)=\frac{d_F T}{2\pi}\log T + \frac{T}{2\pi }\log (\lambda Q^2)-\frac{d_F T}{2\pi} + O(\log T),\N\]\Nwhere \(N_k(a, T, F)\) denotes the number of \(a\)-points of \(F^{(k)}(s)\) in the region \(0 < t < T\) and \(E_2 < \sigma < E_1\), where \(E_1\) and \(E_2\) are described explicitly in the paper. Note that the degree \(d_F\) and the parameter \(\lambda\) are arithmetic invariants that can be computed from the functional equation of \(F(s)\).\N\NFinally, the authors establish an asymptotic formula for \N\[\N\sum_{\substack{ 0<\gamma_a^{k} < T\\ E_1<\beta_a^{k} < E_2}} x^{\rho_a^{(k)}}\quad\text{as }T\to\infty,\N\]\Nwhere \(x\) is a positive real number such that \(x>1\) and \(\rho_a^{(k)} = \beta_a^{(k)} + i\gamma_a^{(k)}\) denotes an \(a\)-point of \(F^{(k)}(s)\).