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

Hung Bui, Richard R. Hall

Research output: Contribution to journalArticlepeer-review


Let Z(k)(t) be the k-th derivative of Hardy’s Z-function. The numerics seem to suggest that if k and ℓ have the same parity, then the zeros of Z(k)(t) and Z(ℓ)(t) come in pairs which are very close to each other. That is to say that Z(k)(t)Z(ℓ)(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(ℓ)(t)Z(m)(t)Z(n)(t).
Original languageEnglish
Pages (from-to)2304–2323
JournalBulletin of the London Mathematical Society
Early online date16 May 2023
Publication statusPublished - 2 Oct 2023


  • Riemann zeta-function
  • Hardy’s Z-function
  • moments


Dive into the research topics of 'On the derivatives of Hardy's function Z(t)'. Together they form a unique fingerprint.

Cite this