On the limitations of Li's criterion for Riemann Hypothesis

The Li criterion reformulates the Riemann Hypothesis as the positivity of a sequence of real coefficients associated with the Riemann xi function, suggesting a potentially more tractable alternative to the geometric analysis of zeta zeros. This has led to sustained efforts to control the oscillatory behavior of these coefficients, often by estimating auxiliary quantities derived from local expansions of $\zeta(s)$ and its logarithmic derivative. We examine these approaches from a structural perspective. Using the Bombieri–Lagarias decomposition of the Li coefficients, we analyze in detail strategies that seek to bound the oscillatory part via estimates of the associated $\eta$–coefficients, including approaches based on Newton–Girard type identities. While these methods clarify structural relationships and yield partial information, they fail to provide effective control of the oscillatory component. We argue that this failure is not accidental but intrinsic: any such estimation scheme necessarily reintroduces zero information in a disguised form. This self-referential feature places fundamental limitations on the Li criterion as a practical tool for establishing the Riemann Hypothesis, despite its formal equivalence.