Order of an entire function (${\rho }_f={\mathrm{limsup}}_{R\to \infty }\log \log M_f(R)/\log R$)

For a nonconstant entire function, put $M_f(R)={\max }_{|s|\le R}|f(s)|$. Its order measures the exponent in its exponential growth. Bounds $\log M_f(R)=O(R\log R)$ imply order at most one; a matching lower bound along the positive real axis proves order exactly one. This distinction is stronger than mere entire analyticity and controls the genus in Hadamard factorization.

// Target: analytic-number-theory.bigb