Add three conjectures from Farey spectroscopy research

[SaarShai]

Summary

Three conjectures arising from the study of per-step Farey discrepancy and its connection to Riemann zeta zeros:

1. Dirichlet Polynomial Avoidance Conjecture (novel)

For fixed K ≥ 2, the truncated Möbius Dirichlet polynomial c_K(s) = Σ_{k=2}^K μ(k)k^{-s} is nonzero at every nontrivial zeta zero. Verified by rigorous interval arithmetic for K ∈ {10, 20, 50} at the first 100 zeros (300/300 cases certified). The polynomial has infinitely many zeros in the critical strip (Langer 1931), but they systematically avoid zeta zero ordinates.

2. Farey Bridge Identity

The exponential sum over F_{p-1} at frequency p equals M(p) + 2, connecting Farey geometry to the Mertens function via Ramanujan sums. Substantially formalized with one remaining sorry in the assembly step.

3. Farey Sign Pattern (density-one)

sgn(ΔW(p)) = sgn(-M(p)) for a density-one subset of primes, where ΔW is the change in Farey Weyl discrepancy. Universal version disproved at p = 243,799. Verified for all 4,617 primes ≤ 100,000 with M(p) ≤ -3.

Source

• Paper: "Prime Spectroscopy of Riemann Zeros" by Saar Shai (2026)
• Code + 434 Lean 4 verified results: https://github.com/SaarShai/Primes-Equispaced
• AI Disclosure: Conjectures formulated with assistance from Claude (Anthropic)

AMS Classification

All three fall under AMS 11 (Number Theory), primarily 11M26 (Riemann zeta zeros) and 11B57 (Farey sequences).

Notes

• The DPAC is the most novel conjecture — no prior literature states this precisely
• The Bridge Identity is close to proved (all lemmas done, one assembly sorry)
• The Sign Pattern has a known counterexample boundary, making it a sharp target for formalization

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