The Riemann Zeta Function on Vertical Arithmetic Progressions

We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression ${1 \over 2} + {\rm i}(an+b)$ with a>0, b real, exhibits a remarkable correspondence with the analogous continuous average and derive several consequences. For example, motivated by the linear independence conjecture, we show at least one third of the elements in the arithmetic progression an+b are not the ordinates of some zero of ζ(s) lying on the critical line. This improves on an earlier work of Martin and Ng. We then complement this result by producing large values of ζ(s) on arithmetic progressions which are of the same quality as the best Ω results currently known for $\zeta({1 \over 2}+{\rm i}t)$ with t real.