Spira’s RH for sections of the Hardy Z-function

Prove that all non-trivial zeros of the high-order section Z_{N(t)}(t) = \sum_{k=1}^{N(t)} \cos(\theta(t) - \ln(k) t)/\sqrt{k} of the Hardy Z-function, with N(t) = \lfloor t/2 \rfloor and \theta(t) the Riemann–Siegel theta function, lie on the real axis.

Background

The paper studies two approximations to the Hardy Z-function: the classical Hardy–Littlewood approximate functional equation using a main sum up to $\widetilde{N}(t) = \lfloor \sqrt{t/(2\pi)} \rfloor$, and Spira’s higher-order section approximation using the sum up to $N(t) = \lfloor t/2 \rfloor$. Spira observed empirically that his approximation appears to have all zeros on the real axis, in contrast to the classical AFE which requires further refinement via the Riemann–Siegel expansion to control zeros.

To formalize Spira’s observation, the authors state a precise conjecture asserting that all non-trivial zeros of the section $Z_{N(t)}(t)$ are real. They then provide theoretical justification via accelerated approximations suggesting that Spira’s conjecture is essentially equivalent to the Riemann Hypothesis for $Z(t)$.