Derivative Of Riemann Zeta Function Research Articles

Zeros of higher derivatives of Riemann zeta function

In this article, we extend the result of Conrey [4, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of ζ(k). Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's ηk-function cluster near the critical line.

Zero-free regions of derivatives of Riemann zeta function

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.