May 15, 2003. Peter Zvengrowski (University of Calgary, Canada). A Horizontal Approach to the Riemann Zeta Function

This talk will start by introducing the Riemann zeta function and tracing some of its historical development, along with the closely related question of the distribution of the primes. We shall then discuss some recent work by Filip Saidah and the author, which compares |\zeta(1/2) - \Delta + it)| and |\zeta(1/2) + \Delta + it)|, i.e. the moduli of \zeta at two points symmetrically located with respect to the critical line \sigma = 1/2. Very accurate bounds, both upper and lower, are obtained for the ratio of these two quantities. We also prove that for t>2\pi+1, 0 \le \Delta \le 1/2, |\zeta(1/2) - \Delta + it)| > |\zeta(1/2) + \Delta + it)|, and remark that strengthening this to a strict inequality (for \Delta > 0) implies the Riemann hypothesis.

List of talks at previous sessions of the seminar.