According to Lindelöf's theorem, given points $z_i\in \mathbb C\setminus \{0\}$ ordered by increasing modulus with possible repetitions, we can define a function
$$
f(z)=\prod_{n=1}^\infty (1-z/z_n)e^{z/z_n}$$
which is of exponential type iff the number of $z_i$ in the ball $B(0,R)$ is bounded by $cR$ for some constant $C>0$ and
$$\sup_{R>0}\sum_i z_i^{-1}<\infty,$$
(see the book of Koosis, *The Logarithmic Integral*).
My question is whether, under the same conditions, by adapting the definition of $f$, we can also prescribe the type, i.e. the infimum of the $a>0$ such that $$\sup_z |f(z)|\exp(-a|z|)<\infty.$$

As a step in this direction, I believe I managed to do it under the condition $\sum_i |z_i|^{-1}<\infty$, by defining $$f(z)=\prod_n \frac{s\left(\frac{z_n-z}{\lambda z_n}\right)}{s(\lambda^{-1})}$$ with the right choice of the function $s$ and the parameter $\lambda>0$ chosen sufficiently large (one can show that the spectrum of $f$ is bounded by $a=K\lambda^{-1}\sum_i |z_i|^{-1}$ for some constant $K$ depending on $s$, which means the type is smaller than $a$ by the Schwartz–Paley–Wiener Theorem).

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