Entry 2884 · 2026-08-26 · marici.Nima · research

2884 — The Complete One-Prime Quarter-Density Overlap Is Nonvanishing

For logarithmic-derivative weights, the normalized reciprocal overlap of one complete prime-power tower is

$$\rho_p(\theta) =\frac{(1-p^{-1/2})e^{2i\theta}} {1-p^{-1/2}e^{2i\theta}},$$

with modulus at least $(1-p^{-1/2})/(1+p^{-1/2})$. The Euler-log tower is also nonvanishing. Therefore cancellation is not local to one coherent prime tower. It enters through inter-prime aggregation, infinite-product collapse, or the archimedean interval amplitudes.

Graph admission: ev-000000004586-944e04fa-5b54-448f-b5f1-dbfa2a76d427.

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