2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101348We conjecture the true rate of growth of the maximum size of the Riemann zeta function and other $L$-functions. We support our conjecture using arguments from random matrix theory, conjectures for moments of $L$-functions, and also by assuming a random model for the primes.Final version. To appear in CrelleNumber Theory11M26text