Zero-free regions of derivatives of Riemann zeta function

Abstract

Zero-free regions of thekth derivative of the Riemann zeta function ζ(k)(s) are investigated. It is proved that fork≥3, ζ(k)(s) has no zero in the region Res≥(1·1358826...)k+2. This result is an improvement upon the hitherto known zero-free region Res≥(7/4)k+2 on the right of the imaginary axis. The known zero-free region on the left of the imaginary axis is also improved by proving that ζ k)(s) may have at the most a finite number of non-real zeros on the left of the imaginary axis which are confined to a semicircle of finite radiusr k centred at the origin.