Correction to the first printing of third edition: The exponent in the result should be \(-3 / 2\). After the density of electrons levels off at the nearly the proton density, you can use equation (9.5.8) to show that the chemical potential \(\mu_{-} \approx m c^{2}\). Then the positron number density is given by equation (9.5.7),

\[
\begin{aligned}
\frac{n_{+}}{n_{\gamma}} & \approx \frac{1}{\zeta(3)} \int_{\beta m c^{2}}^{\infty} \frac{x \sqrt{x+\beta m c^{2}} \sqrt{x-\beta m c^{2}} d x}{e^{x+\beta m c^{2}+1}} \\
& \approx \frac{e^{-2 \beta m c^{2}}\left(\beta m c^{2} \right)^{3 / 2}}{2 \zeta(3)} \int_{0}^{\infty} \sqrt{y} e^{-y} d y \\
& \approx \frac{e^{-2 \beta m c^{2}}\left(\beta m c^{2} \right)^{3 / 2} \sqrt{\pi}}{4 \zeta(3)} .
\end{aligned}
\]