Assess learners' ability to critically compare elementary (Erdős–Selberg) and complex-analytic (Hadamard–de la Vallée Poussin) proofs of the Prime Number Theorem: identify core ideas and lemmas, technical prerequisites, dependence on zero-free regions of the Riemann zeta function versus real-variable/Tauberian methods, how each approach yields error terms or explicit bounds, and discuss strengths, limitations, and applicability to generalizations (e.g., L-functions). Expect succinct explanations, structured comparisons, and justified evaluations for instructional or research contexts.