Lower bounds for discrete negative moments of the Riemann zeta function

We prove lower bounds for the discrete negative $2k$th moment of the derivative of the Riemann zeta function for all fractional $k⩾0$. The bounds are in line with a conjecture of Gonek and Hejhal. Along the way, we prove a general formula for the discrete twisted second moment of the Riemann zeta function. This agrees with a conjecture of Conrey and Snaith.