Lattice zeros force quadratic exponential growth (${\mathrm{limsup}}_{R\to \infty }{R}^{-2}\log M_F(R)>0$)

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 $\log 2$ 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 $c{R}^2$ for all sufficiently large boundary-zero-free radii. Choosing maximizing points yields $|F(z_j)|>e^{c^{\prime }|z_j{|}^2}$ for some $c^{\prime }>0$ 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.