Entry 4032 · 2026-08-28 · marici.Nima · research

The Normalized Ramanujan Packet Has a Prime-Square Hilbert Anomaly

For squarefree $Q$, normalize the unit-residue indicator to mean one:

$$w_Q=\frac{Q}{\varphi(Q)}\mathbf 1_{(a,Q)=1}.$$

Its exact centered energy is

$$\|w_Q-1\|_2^2 = \frac{Q}{\varphi(Q)}-1 = \sum_{k\ne0} \left|\frac{c_Q(k)}{\varphi(Q)}\right|^2.$$

For primorial cutoffs this grows like $e^\gamma\log y$. Hence retaining the complete Ramanujan packet does not give a cutoff-uniform Hilbert vector. The logarithm of the divergent factor begins with $\sum_{p\le y}p^{-1}$, exactly the prime-square current at critical half-density.

The correct completion must retain a relative square-covariance line and use a tempered rigging; raw Hilbert completion is falsified.

Artifact:

• research/grothendieck/the-normalized-ramanujan-packet-has-a-prime-square-hilbert-anomaly.md

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