Zeros of higher derivatives of Riemann zeta function

Mithun Kumar Das, Sudhir Pujahari

Published 2021-04-20Version 1

In this article, we refined the error term of some results of Levinson and Montgomery \cite{LM}, Ki and Lee \cite{KL} on zero density estimates of $\zeta^{(k)}$. We also have shown that all most all zeros of Matsumoto-Tanigawa's $\eta_k$-function are near the critical line $\frac{1}{2}$. Moreover, we obtain an asymptotic expression for the mean square of the product of higher derivatives of Hardy's $Z$-function with Dirichlet polynomials whose $n$-th coefficient is arbitrary and of size $\ll n^{\epsilon}$, for $\epsilon >0$. For $k=0$ our result recovers the result of Balasubramanian, Conrey and Heath-Brown \cite{BCH} %and also we for shorter intervals with relatively small mollifier length. More generally, we consider the mean square of the product of two arbitrary order derivatives of Hardy $Z$-function with Dirichlet polynomials and by using this we obtain a similar type of mean square result for the product of two arbitrary order derivatives of the Riemann zeta function with mollifiers.