= Lattice zeros force quadratic exponential growth
{title2=$\limsup_{R\to\infty}R^{-2}\log M_F(R)>0$}
A nonzero <entire function> vanishing at every nonzero point of the integer lattice has at least a constant times $R^2$ zeros in the disc of radius $R/2$. Each contributes at least $\log2$ to <Jensen's formula> at radius $R$. Therefore the circle mean of $\log|F|$, and hence the logarithm of its maximum modulus, is bounded below by $cR^2$ for all sufficiently large boundary-zero-free radii. Choosing maximizing points yields $|F(z_j)|>e^{c'|z_j|^2}$ for some $c'>0$ and $|z_j|\to\infty$. If zero is also a zero, first factor out its finite multiplicity; the additional $m\log R$ term does not weaken the quadratic lower bound.
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