Category: Number Theory

Summary: Testing whether intermittently resetting the random sign assignments in a multiplicative function creates a nonmonotonic window of greater partial-sum stability.

Random multiplicative functions are a probabilistic model for studying cancellation and fluctuation in number-theoretic sums. This experiment asks what happens when the random sign assignments attached to primes are not fixed forever, but are instead reset intermittently on a controllable schedule.

The simulation generates partial sums under several reset intervals and compares how large the excursions become, how often the sign flips, and how long the sums dwell on one side of zero. The core question is whether moderate resetting suppresses extreme excursions better than either never resetting or resetting too aggressively.

That is interesting because it introduces a temporal control parameter into a standard random-multiplicative setting. Rather than only asking how large the fluctuations are, the experiment asks whether there is an interior regime where instability is reduced most effectively.

Method: Repeated simulations of random multiplicative partial sums under different reset intervals, measuring excursion size, sign persistence, and sign-change statistics.

What is measured: Maximum absolute partial sum, terminal absolute partial sum, sign changes, longest sign dwell, best reset interval, and gains of medium resetting relative to no resetting.