No abstract

Dedicated to Hugh Montgomery on his 60th birthday.Abstract. We conjecture the true rate of growth of the maximum size of the Riemann zetafunction and other L-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of L-functions, and also by assuming a random model for the primes.

We use a smoothed version of the explicit formula to find an approximation to the Riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes. We model the first product by characteristic polynomials of random matrices. This provides a statistical model of the zeta function that involves the primes in a natural way. We then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line. This calculation illuminates recent conjectures for these moments based on connections with random matrix theory.

Random matrix theory is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ± (s), evaluated at the complex zeros 1 2 + i® n . We also discuss the probability distribution of ln j± 0 (1=2 + i® n )j, proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations.