Besfort Shala

University of Bristol University of Bristol


Central limit theorems for random multiplicative functions


Probability Seminar


18th October 2024, 3:30 pm – 4:30 pm
Fry Building, 2.04


Sums of multiplicative functions are of central importance in number theory, as they carry deep arithmetic information about the distribution of primes. For instance, the Riemann Hypothesis is equivalent to obtaining a law of iterated logarithm type upper bound for partial sums of the Möbius function. Being far from proving the Riemann Hypothesis, number theorists often resort to probabilistic heuristics and settings where we can prove random analogues of results that we expect to be true deterministically. As such, it becomes interesting to study partial sums of random models of multiplicative functions. Even though we know a lot about the latter due to work of Harper, the exact distribution remains elusive.

In this talk, I will focus on restricted partial sums, for which it may be possible to understand the distribution and show that it converges to a standard Gaussian when appropriately normalized. There are several results in this direction, but of particular number-theoretic importance is evaluating random multiplicative functions on polynomial arguments. On the deterministic side, this corresponds to understanding matters about primes that lie beyond the Riemann Hypothesis, such as the twin prime conjecture. I will explain the work of Klurman-Shkredov-Xu (in the case when the distribution of the random multiplicative function on the primes is uniform on the complex unit circle), and joint work of mine with Jake Chinis (in the case when the distribution of the random multiplicative function on the primes is Rademacher), where we establish convergence in distribution to a standard Gaussian and almost sure law of iterated logarithm type lower bounds for partial sums of random multiplicative functions evaluated on polynomial arguments. I will focus on the probabilistic aspects of our work, where we utilize tools from martingale theory and conditional multivariate Gaussian approximation. If time permits, I will talk about some related problems that remain open, some of which we address in ongoing work with Christopher Atherfold.






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