Simplified Equation for the Bare Charge in Renormalized Quantum Electrodynamics

The bare coupling constant of quantum electrodynamics, if finite, is the zero of the Gell-Mann-Low function $\psi$. In the paper to which this note is being appended, it was shown that the zeros of a function $f$ are also zeros of $\psi$. The quantity $f$ is the sum of the coefficients of the single power of the logarithm of a cutoff present in every order of perturbation theory in the vacuum polarization, omitting all those diagrams which include insertions in single internal photon lines. In this note the converse is shown; the zeros of $\psi$ are also zeros of $f$. It is further shown that the zeros of $\psi$ are zeros of a still simpler function $f_1$, which is defined analogously to $f$, but with the exclusion of all diagrams which contain multiple closed electron loops.