The Lindeberg central limit theorem (CLT) is a fundamental result in probability and statistics, providing sharp conditions under which sums of independent random variables converge to a normal distribution. In this talk, we first review the historical development of the Lindeberg CLT and the accompanying Lindeberg method, a technique based on smooth function approximations that has become a powerful tool for proving limit theorems beyond classical settings.

We then survey recent advances in Stein’s method, a versatile approach to distributional approximation. Our focus includes both normal and nonnormal approximations, highlighting improvements in error bounds and convergence rates. Finally, we briefly discuss a surprising connection between Stein’s method and the Riemann hypothesis, offering a probabilistic perspective on a central problem in analytic number theory.