Adaptation of the Durkan–Pearce-Crump method to obtain the stronger bounds

Determine whether the method of Durkan and Pearce-Crump for proving Harper’s conjecture can be adapted to recover the stronger family of almost-sure upper bounds for partial sums of Steinhaus and Rademacher random multiplicative functions established by Theorem 1, namely bounds of the form \(\sqrt{x}(\log_2 x)^{1/4}G(\log_3 x)\) under the stated summability condition on \(G\).

Background

The paper proves Harper’s conjecture on the almost-sure upper bound for partial sums of Steinhaus and Rademacher random multiplicative functions, and in fact obtains a more precise family of bounds through a critical-chaos and stopped-martingale argument. Durkan and Pearce-Crump independently proved Harper’s conjecture using a different approach involving stopping, higher-moment estimates, a finer prime-block decomposition, a deterministically tilted supermartingale, scalar maximal inequalities, hypercontractivity, and divisor weights.

The authors explicitly leave unresolved whether the Durkan–Pearce-Crump approach can be modified to yield the stronger conclusion of their own Theorem 1, rather than only the conjectured exponent of $\log_2 x$.