CA2 numbers and the Riemann hypothesis: a numerical investigation of Ramanujan's inequalities

Abstract

Robin's criterion reduces the Riemann hypothesis to an inequality for colossally abundant (CA) numbers, and Wu's theorem reduces the test further to the subclass CA2. We study this reduction numerically and consider smaller subsets on which the same criterion remains sufficient. We also show that the logarithmic decay law suggested by Briggs's computations follows from Ramanujan's conditional bounds. To investigate the next term, we compare the normalized Robin difference with Ramanujan's oscillatory expression involving the nontrivial zeros of the zeta function. The resulting difference has a slowly varying component on both CA and CA2. Its scaled values continue to vary over the computed range, leading to questions about convergence, upper bounds, and higher-order terms. We describe the sampling used in the figures and estimate the error from truncating the sum over zeros.