Lower bounds for discrete negative moments of the Riemann zeta function

We prove lower bounds for the discrete negative 2kth 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.