On the Interaction of Li Coefficients and Zeta Zeros in a Poisson-Jensen-Nevanlinna-Framework

We analyze the function $f(z) = \frac{z}{1-z}\zeta\left(\frac{1}{1-z}\right), |z|<1,$ by means of the Poisson–Jensen–Nevanlinna theorem. Zeros of $f$ inside the unit disk correspond to nontrivial zeros $\rho$ of the Riemann zeta function off the critical line, mapped by $z = 1-1/\rho$. We show that the local logarithmic Taylor coefficients of $f$, which encode the oscillatory components of the Li coefficients, enter the Poisson–Jensen– Nevanlinna expansion on equal footing with explicit discrete contributions from the zeros. As the radius $r$ increases, contributions that appear divergent when viewed solely through the Taylor (Li-type) coefficients are systematically matched the zero terms once $r$ exceeds the modulus of the corresponding zeros. This matching mechanism prevents any net divergence of the Poisson–Jensen–Nevanlinna coefficients in the hypothetical presence of zeros off the critical line. In particular, both the analytic identity and the convergence of the full expansion are preserved, and no uncancelled growth remains that could be exploited to derive a contradiction. Our analysis demonstrates that approaches to the Riemann Hypothesis based exclusively on the behavior of Li-type coefficients overlook a structural compensation inherent in the Poisson–Jensen–Nevanlinna framework. The full higher-order Jensen structure reveals a built-in stability mechanism that obstructs divergence-based arguments and clarifies the limitations of coefficient-based criteria when detached from the global zero distribution.