Equivalence of Definitions of Absolute Convergence of Product of Complex Numbers

Theorem
Let $(z_n)$ be a sequence of complex numbers with real part $>-1$.

Then the following are equivalent:


 * The infinite product $\displaystyle\prod_{n=1}^\infty(1+z_n)$ converges absolutely.
 * The series $\displaystyle\sum_{n=1}^\infty\log(1+|z_n|)$ converges absolutely.
 * The series $\displaystyle\sum_{n=1}^\infty\log(1+z_n)$ converges absolutely.
 * The series $\displaystyle\sum_{n=1}^\infty z_n$ converges absolutely.

Proof
By Convergence of Infinite Product of Complex Numbers, the first two are equivalent.

Because $\log z\asymp z$ for $|z|<1-\delta$, the last three are equivalent.