Non-Markovian memory in the fluctuations of the sequence of prime gaps

Abstract

This paper utilizes a white noise stochastic framework using a functional integral approach to model the inherent fluctuations of the sequence of prime gaps by fitting its mean square deviation (MSD) to a theoretical MSD with the form of an exponential decay containing a memory parameter b. Sequences of prime gaps of varying orders of magnitude up to N = 10¹⁰ were generated using a prime sieve package in Python. The normalized sequences of the prime gaps were modeled as the inherent fluctuations of the sequences of prime gaps. Results suggest that these fluctuations have Non-Markovian memory that weakens linearly with log₁₀ N due to a nonzero memory parameter b which would also suggest that neighboring normalized prime gaps are correlated. The negative linear relation between the memory parameter b and log₁₀ N, provided that this trend goes on beyond 10¹⁰, implies that the fluctuations of the sequence of prime gaps eventually become memoryless at very large values of N which may be used as evidence of the randomness of very large prime gaps.