= Solution
Let $\delta_{\rm sc}\simeq1.686$ be the <linear spherical-collapse threshold> and let $D(t)$ be the <linear growth factor>, normalized to one at the epoch to which the random density field has been extrapolated. The time-dependent barrier is $\delta_c(t)=\delta_{\rm sc}/D(t)$. The <smoothed matter density variance> $\sigma^2(M)$ is the variance of the linear <density contrast> smoothed with a <spherical top-hat window function> whose comoving mass is $M$. Both quantities must refer to the same linear normalization; equivalently use $\nu=\delta_{\rm sc}/\sigma(M,t)$.
Let $N$ denote the comoving number density and write $\mathcal N(M,t)=dN/dM$ for the positive differential <Press-Schechter halo mass function>. With $\bar\rho_0$ the mean comoving mass density, the mass fraction obeys
$$
f(>M,t)=\frac1{\bar\rho_0}\int_M^\infty M'\mathcal N(M',t)\,dM',\qquad \mathcal N(M,t)=-\frac{\bar\rho_0}M\frac{\partial f}{\partial M}.
$$
Differentiating the <complementary error function> and using $\nu=\delta_c/\sigma$ gives
$$
\boxed{\frac{dN}{dM}=\sqrt{\frac2\pi}\frac{\bar\rho_0}{M^2}\nu e^{-\nu^2/2}\left|\frac{d\ln\sigma}{d\ln M}\right|.}
$$
This is the requested mass function; $M\,dN/dM$ is the abundance per unit natural logarithmic mass. For physical rather than <comoving number density>, multiply by $(1+z)^3$. The normalization in the <Press-Schechter formalism> includes its factor of two relative to the positive Gaussian tail, so $f(>M)$ tends to one as the smoothing mass tends to zero when $\sigma$ diverges. It is a collapse prescription, not the probability that an arbitrary already nonlinear density field stays Gaussian.
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