  We obtain asymptotic formulae for the second discrete moments of the Riemann
zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. 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 $e^{\pi k/a}$ for
positive integers $k$.
