Mollifiers, long and short

This is joint with David Farmer, Chung Hang Kwan, Yongxiao Lin, and Caroline Turnage-Butterbaugh.
When studying the zeros of Riemann zeta function at a height $T$ up the critical strip one often multiplies
$\zeta$ times a Dirichlet polynomial, called a mollifier, of length $T^{\theta}$ before averaging in order to
pacify the irregularities of $\zeta$. The $\theta$ parameter here is critical.
Farmer conjectured that the mean square formulas one obtains for mollified zeta for small $\theta$ actually
hold for all $\theta$ (the $\theta=\infty$ conjecture). Bettin and Gonek proved that the Riemann Hypothesis
follows. It is known that $\theta$ must be at least $1/6$ to deduce information about simple zeros of zeta.
Here we give new information about how small $\theta$ can be and still deduce (from Levinson's method)
that a positive proportion of the zeros of zeta are on the critical line.