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arXiv · 2304.05178

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

Hung M. Bui·R. R. Hall
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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, $Z^{(k)}(t)Z^{(\ell)}(t)Z^{(m)}(t)Z^{(n)}(t)$.

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Hung M. Bui, R. R. Hall. 2023-04-11. On the derivatives of Hardy's function $Z(t)$. https://arxiv.org/abs/2304.05178

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