  We consider a specific family of analytic functions $g_{\alpha,T}(s)$,
satisfying certain functional equations and approximating to linear
combinations of the Riemann zeta-function and its derivatives of the form
  $c_0\zeta(s)+c_1\frac{\zeta'(s)}{\log T}+c_2\frac{\zeta''(s)}{(\log
T)^2}+\dots+c_{K}\frac{\zeta^{(K)}(s)}{(\log T)^{K}}$.
  We also consider specific mollifiers of the form $M(s)D(s)$ for these linear
combinations, where $M(s)$ is the classical mollifier, that is, a short
Dirichlet polynomial for $1/\zeta(s)$, and the Dirichlet polynomial $D(s)$ is
also short but with large and irregular Dirichlet coefficients, and arises from
substitution for $w$, in Runge's complex approximation polynomial for
$f(w)=\frac1{c_0+w}$, of the Selberg approximation for
  $\frac{c_1}{\log T}\frac{\zeta'}{\zeta}(s)+\frac{c_2}{(\log
T)^2}\frac{\zeta''}{\zeta}(s)+\dots+\frac{c_{K}}{(\log
T)^{K}}\frac{\zeta^{(K)}}{\zeta}(s)$
  (analogous to Selberg's classical approximation for
$\frac{\zeta'}{\zeta}(s)$).
  Exploiting the functional equations previously mentioned (concerning
translation of the variable $s$), together with the mean-square asymptotics of
the Levinson-Conrey method and the Selberg approximation theory (with some
additional results) we show that almost all of the zeros of the Riemann
zeta-function are on the critical line.
