Limitations to mollifying $\zeta(s)$

We establish limitations to how well one can mollify the Riemann zeta-function on the critical line with mollifiers of arbitrary length. Our result gives a non-trivial lower bound for the contribution of the off-diagonal terms to mollified moments of \zeta(s). On the Riemann Hypothesis, we establish a connection between the mollified moment and Montgomery's Pair Correlation Function.