Visualizing the Gilbreath Expectation Sequence

Summary

Terry Tao demonstrates how coding agents facilitate converting code-generated figures into interactive applets, using Figure 1 from his recent paper on the Gilbreath conjecture with Chase and Hunter. The figure plots exact and Monte Carlo simulated values of a sequence called the "Gilbreath expectation sequence," which arises from a Gilbreath array: an inverted pyramid with top entries as independent exponential random variables and subsequent entries as absolute differences of the two above. The sequence represents expectations of left diagonal entries. The paper provides theoretical values for some cases, with simulations agreeing well. Michael Ross later extended theoretical values, maintaining the fit. The asymptotic behavior remains mysterious; the sequence is non-monotonic and not proven bounded. The paper establishes an inequality showing the sequence cannot decay faster than a certain rate.