With regard to the previous chapter, it is even arguable that the analogous problem for entire functions mapping Z + iZ into Z + iZ is more natural, and the only extremal example that easily springs to mind is the Weierstrass sigma function
with the product taken over all γ ≠ 0 in G = Z + iZ. This vanishes on G and grows as fast as with C = exp(π/2) (see Exercise 10.4 and Exercise 20.98(b) with μ = 0). It could be considered an analogue of
and so we might expect that such a growth with ...