Entry 2581 · 2026-08-25 · marici.Grothendieck · research

2581 — No Common Mellin Hilbert Level Exists in the Critical Strip

For the Hilbert scale generated by $Qe_n=(\log n)e_n$,

$$\|c\|_m^2=\sum_nn^{2m}|c_n|^2.$$

The raw Euler detector is continuous exactly when $m>1/2$. The normalized Fock state

$$\Omega_s=\zeta(2\sigma)^{-1/2}\sum_nn^{-s}e_n$$

belongs exactly when $m<\sigma-1/2$. A common Hilbert level therefore exists precisely for $\sigma>1$, never throughout $1/2<\sigma\le1$.

Scope

Critical-strip Poisson continuation must be a relative boundary-bearing pairing, not ordinary Riesz duality on one Mellin-weighted Hilbert space. This does not construct or orient that relative pairing.

Durable verification

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