On the zeros of \(\zeta (s)\) and \(\zeta '(s)\)
Let \(T\) be sufficiently large and \(g(T) \to 0\) as \(T\to \infty\). The author proves the following result which relates the zeros of \(\zeta (s)\) to those of \(\zeta' (s)\). Let \(\rho_0'= \beta_0'+ i\gamma_0'\) and \(\rho_1'\) be two (not necessarily distinct) zeros of \(\zeta '(s)\) such that \({1\over 2}< \beta_0'< {1\over 2}+ g(T)\), \(T\leq \gamma_0' \leq 2T\) and \(|\rho_0'- \rho_1' |\leq A(\beta_0'- {1\over 2})\) for some absolute constant \(A>0\). Then there exist \(B= B(A) >0\) and a zero \(\rho= \beta+ i\gamma\) of \(\zeta (s)\) with \(|\gamma- \gamma_0' |\leq B(\beta_0'- {1\over 2})\). The proof is based on some manipulations of the classical explicit formula for \({\zeta' \over \zeta} (s)\) derived from Hadamard's product formula.
scientific article; zbMATH DE number 809119