= Solution
For the unrestricted zero sequence, introduce the <Weierstrass elementary factors>
$$
E_p(u)=(1-u)\exp\left(u+\frac{u^2}{2}+\cdots+\frac{u^p}{p}\right).
$$
For $|u|\le1/2$, cancellation of the first $p$ terms of the logarithm <power series> gives a <holomorphic logarithm> of the factor:
$$
\log E_p(u)=-\sum_{j=p+1}^\infty\frac{u^j}{j},\qquad
|\log E_p(u)|\le2|u|^{p+1}.
$$
Define the <canonical product>
$$
f(z)=\prod_{n=1}^\infty E_n(z/z_n).
$$
Fix any compact disc $|z|\le R$. Since $|z_n|\to\infty$, eventually $|z/z_n|\le1/2$ uniformly on this disc. The tail logarithms are then bounded by $2(1/2)^{n+1}$, a summable bound independent of $z$. Their sum is <holomorphic> by <locally uniform convergence of holomorphic functions>, and its exponential is nowhere zero. Multiplication by the finite initial product therefore gives an <entire function>. Each factor has a <simple zero> exactly at its specified $z_n$, the exponential factor has none, and a tail beginning beyond any given compact disc is nonvanishing there. Thus \b[$f$ has precisely the prescribed zeros and no others], each simple, and $f(0)=1$. This proves the required existence without an assumption on $\sum_n|z_n|^{-1}$.
For the final, unheaded product request, suppose now that $\sum_n|z_n|^{-1}<\infty$. On $|z|\le R$, all sufficiently late factors have $|z/z_n|\le1/2$, so part (a) bounds their <principal complex logarithms> by $2R/|z_n|$. The logarithms converge absolutely and uniformly; <infinite product convergence from logarithmic tails> proves that
$$
P(z)=\prod_{n=1}^\infty(1-z/z_n)
$$
is <entire> and again has exactly the prescribed simple zeros. For its finite partial products,
$$
\prod_{n=1}^N|1-z/z_n|
\le\prod_{n=1}^N(1+|z|/|z_n|)
\le\exp\left(|z|\sum_{n=1}^N|z_n|^{-1}\right).
$$
Passing to the limit proves
$$
\boxed{|P(z)|\le e^{C|z|},\qquad C=\sum_{n=1}^\infty|z_n|^{-1}<\infty.}
$$
In particular $P$ is an <entire function of exponential type>. This argument uses a valid modulus estimate and does not require the false logarithm assertion in part (c).
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