Simple Zeros and Extreme Values of the Riemann Zeta-Function on the Critical Line with Respect to the Zero-Spacing

Recently, Hall proved upper estimates for the second and fourth moment of the extreme values of the Riemann zeta-function between its zeros on the critical line with respect to the zero-spacing. We give a rather simple proof of these estimates; however, our implicit constants are slightly weaker than Hall’s. Further, we consider discrete moments of the first derivative of the zeta-function at its simple zeros on the critical line with respect to the spacing of zeros and prove upper bounds, comparable to Hall’s estimates for extreme values. Finally, we give for both cases upper estimates for higher moments conditional on a conjecture on mean-values of Hardy’s Z-function. The obtained results may be regarded as numerical evidence for Montgomery’s pair correlation conjecture.