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 \(\frac{1}{2} + i(a n + b)\). 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.