Mean-square values of the Riemann zeta function on arithmetic progressions

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2}+i(an+b)$. It reveals noticeable relation between the discrete moments and the continuous moment of the Riemann zeta function. Especially, when $a$ is a positive integer, main terms of the formula are equal to those for the continuous mean value. The proof requires the rational approximation of $e^{\pi k\mathrm{/}a}$ for positive integers $k$.