Mathematical Research Seminar - Archive

I will give a general overview of recent developments in number theory, with many old and new results being proved through the lens of multiplicative functions. Many deep unsolved problems about primes such as the Riemann Hypothesis and the Twin Prime Conjecture are intimately connected with the Möbius function. Viewing the latter as simply one instance of a multiplicative function satisfying certain properties and developing a general theory of multiplicative functions has been very fruitful – most classical results in analytic number theory have been reproved in this framework, without appealing to the analytic continuation of the Riemann zeta function (or more general L-functions). In particular, one may adapt a probabilistic viewpoint and consider so called random multiplicative functions taking the values +1 and -1 on the primes randomly. This was initiated by Wintner, who proved the analogue of the Riemann Hypothesis in this simplified setting. Later this was further developed by Harper, who considered finer distributional questions. If time permits, I will present results on the Twin Prime analogue in this probabilistic setting, based on ongoing joint work with Jake Chinis.