By considering the dynamical system \dot{s} = \zeta(s) it was discovered, then proved, that the contours \Im \Lambda(s) = 0 of the function which satisfies \zeta(1-s) = \Lambda(s)\zeta(s) cross the critical strip on lines which are almost horizontal and straight, and cut the critical line alternately at Gram points and points where \zeta(s) is imaginary. The real part of \zeta(s), when averaged in a modified manner, for fixed values of \sigma over the values on the ``Gram lines'', satisfies a relation which extends a theorem of Titchmarsh giving the average of \zeta(s) over the Gram points as 2, to the entire right hand side of the critical strip.
This talk will be accessible to non-specialists.
Kevin Broughan