  We show that the twisted second moments of the Riemann zeta function averaged
over the arithmetic progression $1/2 + i(an + b)$ with $a > 0$, $b$ real,
exhibits a remarkable correspondance 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 $a n + b$ are not the ordinates of some zero of $\zeta(s)$ lying on
the critical line. This improves on earlier work of Martin and Ng. We then
complement this result by producing large and small values of $\zeta(s)$ on
arithmetic progressions which are of the same quality as the best $\Omega$
results currently known for $\zeta(1/2 + it)$ with $t$ real.
