On the derivatives of Hardy's function $Z(t)$

Abstract

Let $Z^{(k)}(t)$ be the $k$-th derivative of Hardy's $Z$-function. The numerics seem to suggest that if $k$ and $\ell$ have the same parity, then the zeros of $Z^{(k)}(t)$ and $Z^{(\ell)}(t)$ come in pairs which are very close to each other. That is to say that $Z^{(k)}(t)Z^{(\ell)}(t)$ has constant sign for the majority, if not almost all, of values $t$. In this paper we show that this is true a positive proportion of times. We also study the sign of the product of four derivatives of Hardy's function, $Z^{(k)}(t)Z^{(\ell)}(t)Z^{(m)}(t)Z^{(n)}(t)$.