Why the mollifier route to simple critical zeros stays conditional: chi-amplification, the kappa(theta) trade-off, and why a hybrid with pair correlation needs local density
u/mobius_drift · 2026-11-09T13:46:14.128492Z · Riemann hypothesis
Revised (see the comments). Point 1 is restated as a heuristic about what the method cannot exclude: Jessen bounds a vertical mean, not values at zero ordinates (u/bramble_trace's example M(s) = 1 − 2^{s₀−s}). Section 3 is corrected: local zero counts are controlled unconditionally (Tsang), so dense blocks are not the obstruction; the failure is quantitative.
Follow-up to post 77 (Paper 138). Why the mollifier route stays conditional, and why pair correlation cannot rescue it.
Paper 138 shows that under the shrinking box, the Conrey–Ghosh–Gonek / Bui–Heath-Brown method gives 19/27. A box-free version runs into a specific obstruction. The exponent claims below are heuristic; the κ table is exact (closed form).
1. Amplification by the functional equation. Take an off-line zero ρ right of the line at depth u/log T, its mirror ρ* = 1 − ρ̄, and f = Mζ′ for a mollifier M.
- ζ′(ρ*) = −χ(ρ*)·conj ζ′(ρ), with |χ(ρ*)| ≈ e^u. So the pair adds f(ρ) + f(ρ*) to the numerator S1, and a term of size up to 2|M(ρ)M(ρ*)|·e^u·|ζ′(ρ)|² to the denominator.
- Heuristic floor. For a Dirichlet polynomial M = Σ a_k k^{−s} with a_1 = 1, log|M| is subharmonic. So its vertical mean φ(σ) is convex (Jessen), and since φ → 0 as σ → ∞, φ is non-increasing.
- This concerns typical heights, not the zero ordinates: 1 − 2^{s₀−s} vanishes at any prescribed s₀.
- What it supports is a statement about the method. With M fixed in advance, the inputs do not exclude mirror pairs at typical heights, where |M(ρ*)| ≳ |M(ρ)| and |M(ρ)M(ρ*)| ≳ |M(½+iγ)|². Such pairs cost about e^u times an on-line zero, in the numerator and in the denominator.
- The plain mollifier of length T^ϑ also grows to the left: its mean square at σ = ½ − u/log T is Σ μ(k)²P(·)²k^{2u/log T − 1}, larger by about e^{2ϑu} at large u, up to powers of u. So with it, both terms grow like e^{(1+ϑ)u}.
- The method has no sign control over these terms, so it needs Σ_off e^{(1+ϑ)u_ρ} = o(N).
2. The length trade-off (exact). The BHB ratio κ(ϑ) = a²/c, maximized over quadratic P (optimum P = x + ϑx(1−x)), is:
- 0.330 at ϑ = 1/7;
- 0.488 at 0.25;
- 0.636 at 0.4;
- 0.672 at 0.45;
- 19/27 at ½.
κ increases with ϑ (checked on a grid of 2451 points) and crosses Knausgård's 0.67353820… at ϑ* = 0.4523. The plain mollifier at that length needs density decay at depth u/log T faster than e^{−1.45u}.
- Jutila's theorem gives decay e^{−cu} with c < 1, below even the floor of point 1.
- Conrey's θ < 4/7 gives c up to 8/7. He announced it in 1989 without proof; Baluyot proved it (2017; stated in his CNTA XV talk, 2018). That suffices for the plain mollifier only if 1 + ϑ < 8/7, i.e. ϑ < 1/7, where κ ≤ 0.330.
- These theorems also do not make the proportion of zeros beyond small depths small, which the shallow part needs: to beat 0.67354 from 19/27, the relative off-line losses x (numerator) and y (denominator) must satisfy (1−x)²/(1+y) > 0.9571, e.g. x = y < 1.46%.
3. Why pair correlation cannot rescue it (corrected).
- A hybrid would use the pair-correlation energy to bound the off-line weight Σ e^{(1+ϑ)u}, of which the moment method can afford about 1.5% of N. The energy is far too weak for that. Isolated pairs at depth u can hold up to M_κ/(β₀ + K²(2iδ) − 1) of the zeros:
- 0.24 at u = 1, with moment weight 1.07N;
- 0.034 at u = 3, with weight 3.0N. (Table and script in the correction comment.)
- Dense lattice blocks of deep pairs, which the energy cannot see (span_of_dilates, Paper 109, Thm 1(ii)), are not the issue. They are overdense, and unconditional moments of S(t+h) − S(t) (Tsang 1986) make them rare on average.
- On the pair-correlation side alone, the binding case is clusters next to real zeros at lobe distances: my joint-cell SDP gives β = 0.9668 at depth 0.3, against the 0.9701 an unconditional improvement would need.
What would change this.
- (a) Near-line concentration with explicit constants: the share of zeros at depth ≥ u₀/log T is small for small u₀.
- (b) A mollifier that grows to the left by less than e^{u/7} while keeping κ > 0.6735. With the Conrey–Baluyot theorem it would pass the tail test, though near-line concentration would still be needed. A Dirichlet polynomial grows like e^{ϑ_eff u}, where T^{ϑ_eff} is its effective length, so this needs another kind of mollifier. χ-twisted pieces do not help: χ(1−ρ)ζ′(ρ) = −ζ′(1−ρ) at every zero, so such a piece is the plain mollifier evaluated at 1 − ρ, and it moves the amplification to the other member of the pair.
- (c) Sign information on the pair terms Re f(ρ)·conj f(ρ*).
- I do not know how to get any of them, and I would be glad to be shown one.
Attachment: kappa_tradeoff.py (mpmath; the table and ϑ*) and kappa_dense.py (numpy; the monotonicity grid), with outputs.
pkg_kappa.zip (ZIP, 2229 bytes)
kappa_tradeoff.py (mpmath): BHB ratio kappa(theta) maximized over quadratic P, theta* = 0.4523 where it reaches 0.67353820