The statistics of the zeros of the Riemann zeta-function and related topics
This thesis concerns statistical patterns among the zeros of the Riemann zeta function, and conditioned on the Riemann hypothesis proves several related original results. Among these:
By extending a well known result of H. Montgomery, we show, at an only microscopically blurred resolution, that the distance between two randomly selected zeros of the zeta function tends to weakly repel away from the location of low-lying zeros of the zeta function.
For random collections of consecutive zeros that are not so large as to see this resurgence effect, we support the view that they resemble the bulk eigenvalues of a random matrix by in particular proving an analogue of the strong Szego theorem.
Concerning even smaller collections of zeros, we show that a statement that the zeros of the Riemann zeta function locally resemble the eigenvalues of a random matrix (the GUE Conjecture) is logically equivalent to a statement about the distribution of primes. On this basis, we make a conjecture for the covariance in short intervals of integers with fixed numbers of prime factors, weighted by the higher order von Mangoldt function. This is related to the so-called ratio conjecture. The covariance pattern is surprisingly simple to write down.
We finally include a rigorous derivation that uniform variants of the Hardy-Littlewood conjectures agree with the GUE Conjecture. Even thus conditioned, the range of correlation test functions against which we may confirm the GUE pattern for zeta zeros remains limited. We consider in detail the case of two, three, and four point correlations, the two point case being due to Mongtomery.