On a problem connected with zeros of \(\zeta\) (s) on the critical line
The paper deals with estimates of \(\zeta (+it)\) and zeros of \(Z^{(k)}(t)\), where \(Z(t)=\chi^{-}(+it)\zeta (+it),\chi (s)=\zeta (s)/\zeta (1-s).\) The main result, based on the technique of exponential averaging and the theory of exponent pairs, is the following: Let \(k\geq 1\) be a fixed integer. Then every interval \([T, T+H]\) contains a zero of odd order of \(Z^{(k)}(t)\) if \(T\geq T_ 0\) and \(H\gg T^{a(k)} \log T,\) where \[ a(k)=\min (3/37,\quad \mu ()+\epsilon)\;if\;k=1,\quad and \] \[ a(k)=\min (27/(164k+168),\quad (1/k)\mu ()+\epsilon)\;if\;k\geq 2, \] and \(\mu (\sigma)=\limsup_{t\to \infty}\log | \zeta (\sigma +it)| / \log t\). The bounds for a(k) are the sharpest ones hitherto. Bounds for \(\zeta (+iT)\) are also derived, provided that \(\zeta (+it)\) or Z'(t) does not vanish in certain short intervals containing T.
scientific article; zbMATH DE number 4012735