A random-matrix theory of the number sense

A random-matrix theory of the number sense.

Number sense, a spontaneous ability to process approximate numbers, has been documented in human adults, infants and newborns, and many other animals. Species as distant as monkeys and crows exhibit very similar neurons tuned to specific numerosities. How number sense can emerge in the absence of learning or fine tuning is currently unknown. We introduce a random-matrix theory of self-organized...

Random matrix theory and number theory

We review some of the connections between Random Matrix Theory and Number Theory. These include modelling the value distributions of the Riemann zeta-function and other L-functions, and the statistical distribution of their zeros. 1.

Random Matrix Approaches in Number Theory

The connection between random matrix theory and the Riemann zeta function was established in 1973 when Montgomery, who had conjectured the 2-point correlations of the Riemann zeros, and Dyson, who was interested in similar statistics of the eigenvalues of ensembles of unitary matrices, realized that the formulae they had discovered independently were in fact identical in a natural asymptotic li...

Applications of random matrix theory to analytic number theory

This research proposal concerns several projects at the interface of analytic number theory and random matrix theory. As such the proposal involves ideas from number theory, combinatorics, and analysis. The proposal is divided into two sections, related by the perhaps surprising use they make of combinatorial representation theory, but also by the following theme: both sections will study a spe...

A Guide to Random Matrix Theory for Number Theorists