Uniform approximations and zeros of the derivatives of the Hardy \(Z\)- function in short intervals (Q1261254)

Detailed proofs of the results previously announced by the author [Dokl. Akad. Nauk SSSR 307, 28-31 (1989; Zbl 0695.10032)] are given. These include formulas for \(\zeta^{(k)}({1\over 2}+it)\) and \(Z^{(k)}(t)\) which are uniform for \(k\ll\log t\), where \(Z(t)=\chi^{-1/2}({1\over 2}+it)\zeta({1\over 2}+it)\) and \(\chi(s)=\zeta(s)/\zeta(1-s)\). For the latter it is shown that \[ Z^{(k)}(t)=2 \sum_{n\leq P} n^{-1/2} \left(\log {P\over n}\right)^ k \cos\left( t\log{P\over n}-{t\over 2}- {\pi\over 8}+{{\pi k} \over 2}\right)+ O\left(t^{-1/4} \left( {3\over 2}\log t\right)^{k+1}\right) \] uniformly for \(t\geq\max(t_ 0,e^{2k})\), where \(P=(t/2\pi)^{1/2}\). From this it is deduced that, for \(T\geq T_ 0\) and \({{\log T} \over {6\log\log T}}\leq k\leq {{\log T} \over 6}\), every interval \((T,T+9\pi\log T)\) contains a zero of odd order of \(Z^{(k)}(t)\).