Abstract
We study the Robin defect, the difference between the two sides of Robin’s inequality, which is closely connected to the Riemann Hypothesis. We use Ramanujan’s transformation for the divisor Lambert series to write an exact identity for the Laplace transform of this defect. The main point is that positivity after smoothing is not the same as positivity for each integer. This explains why Ramanujan’s transformed identities give strong global control, but cannot alone prove the Riemann Hypothesis. We also show that any possible failure of Robin’s inequality must be checked on special divisor-rich integers, especially colossally abundant and highest abundant numbers. Numerical examples illustrate how the defect behaves for early extremal values. This work separates what Ramanujan’s identities can prove from the pointwise positivity that Robin’s criterion truly requires.
