Berry–Keating conjecture on optimal-order Riemann–Siegel error decay

Determine whether the Riemann–Siegel formula for the Hardy Z-function, when expanded to its optimal order, achieves an exponentially decaying approximation error of order O(e^{-\pi t}).

Background

The Riemann–Siegel formula refines the Hardy–Littlewood approximation by evaluating the error term R(t) to higher orders using saddle-point methods. Although increasing the order does not guarantee improved accuracy for a fixed t, Berry and Keating conjectured that there exists an optimal truncation order yielding very strong accuracy.

The authors cite Berry and Keating’s conjecture that the optimally truncated Riemann–Siegel expansion attains exponentially small error O(e^{-\pi t}). They note their accelerated approximation achieves accuracy consistent with this conjectured benchmark, though the conjecture itself remains open.