Values of the Riemann zeta function on vertical arithmetic progressions in the critical strip

Putnam in 1954 showed that any sequence of consecutive zeros of the Riemann zeta function on the critical line does not form an arithmetic progression. Recently, under a joint work with Junghun Lee, Athanasios Sourmelidis and J\"{o}rn Steuding, we can somewhat extend this result of Putnam for not only zeros, but also sets values of the Riemann zeta function in the critical strip. We could not obtain any results exactly on the critical line, but Lee and I could later show a connection between values of the Riemann zeta function on the critical line and on the right-half of the critical strip. The four of us also considered overlapping of values on the vertical arithmetic progressions. Finally, to attack the problem on the critical line, Steuding and I recently proved an approximate functional equation for the fourth moment of the Riemann zeta function.