= Solution
First fix a genuine normalization error in the original PDF: the Gaussian exponent uses $\sigma^2$, so $\sigma$ must be the <standard deviation>, the <variance> is $\sigma^2$, and the normalized <probability density function> is
$$
p_M(\delta)=\frac1{\sqrt{2\pi}\,\sigma(M,z)}
\exp\!\left(-\frac{\delta^2}{2\sigma^2(M,z)}\right).
$$
The printed prefactor has only $\sigma$ under its square root; its integral is $\sqrt{\sigma}$ rather than one. Use the normalized <normal distribution> above.
The one-scale probability of exceeding the <linear spherical-collapse threshold> is $P_M=\tfrac12\operatorname{erfc}[\delta_c/(\sqrt2\sigma)]$. In the <Press-Schechter formalism>, the mass fraction above $M$ is assigned to
$$
F(>M,z)=2P_M=\operatorname{erfc}\!\left(\frac{\delta_c}{\sqrt2\sigma(M,z)}\right).
$$
The factor two is necessary to recover all the mass in the hierarchical limit $\sigma\to\infty$: the uncorrected one-point upper tail tends to only one half. A precise justification is available in the <excursion-set description of halo formation> with a <sharp-k smoothing filter>. The smoothed <Gaussian random field> then becomes <Brownian motion> in its variance $S$. Reflecting a trajectory after its first crossing of $\delta_c$ pairs trajectories which crossed but end below the threshold with those which end above it. The <Brownian reflection principle> therefore gives $\Pr(\max_{s\le S}\delta(s)>\delta_c)=2\Pr(\delta(S)>\delta_c)$.
This multiscale mass assignment is an extra modelling prescription: a Gaussian one-point probability and a collapse threshold alone do not mathematically determine a halo mass function. The “all matter” normalization additionally assumes arbitrarily large small-scale variance; a finite variance cutoff leaves an uncollapsed fraction.
Differentiate at fixed $z$ and fixed $\delta_c$:
$$
\frac{\partial F}{\partial M}
=\sqrt{\frac2\pi}\frac{\delta_c}{\sigma^2}
\exp\!\left(-\frac{\delta_c^2}{2\sigma^2}\right)\frac{\partial\sigma}{\partial M}.
$$
For decreasing $\sigma(M,z)$, a mass interval contains the positive fraction $-F'(M)\,dM$. Its comoving mass is $\rho_{m,0}[-F'(M)]\,dM$, so division by halo mass gives the <Press-Schechter halo mass function>:
$$
\boxed{\frac{dn}{dM}=\sqrt{\frac2\pi}\frac{\rho_{m,0}}M
\frac{\delta_c}{\sigma^2(M,z)}\left|\frac{\partial\sigma(M,z)}{\partial M}\right|
\exp\!\left(-\frac{\delta_c^2}{2\sigma^2(M,z)}\right).}
$$
The absolute value, or equivalently a minus sign multiplying $\partial_M\sigma$, is essential for a positive abundance. The original PDF correctly places absolute values around the abundance and the full <derivative> expression; these bars are lost in the converted TeX. The <density> is comoving, so the prefactor uses today's matter <mass density>, rather than the physical <density> at redshift $z$.
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