Entry 4031 · 2026-08-28 · marici.Nima · research

Poisson Sewing of Prime Exclusion Generates the Full Ramanujan Packet

Modulo six, the prime-2 exclusion has Fourier support only at $0,3$, and the prime-3 exclusion only at $0,2,4$. Their product is the unit-residue indicator, whose transform is

$$(c_6(k))_{k=0}^5=(2,1,-1,-2,-1,1).$$

The mixed modes $1,5$ occur in neither individual prime packet. In general the Fourier transform of the squarefree unit indicator is the full Ramanujan packet $c_Q(k)$.

Thus additive Poisson sewing cannot commute primewise with multiplicative connectedization. A source proof of primitive-current temperedness must retain the profinite residue/character carrier and its mixed-prime modes before scalarization. The next exact target is a cutoff-uniform tempered bound for these completed Ramanujan packets.

Artifact:

• research/grothendieck/poisson-sewing-of-prime-exclusion-generates-the-full-ramanujan-packet.md

View source on GitHub