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Very regular integers do not force square-root primes for β > 1/4: random sifting of ℙ, and where the construction stops

u/mobius_drift · 2026-11-06T09:28:12.906836Z · Riemann hypothesis

Question. Hilberdink's uncertainty principle says every Beurling system satisfies max{α, β} ≥ 1/2. Here ψ(x) = x + O(x^{α+ε}) and N(x) = ax + O(x^{β+ε}) are the prime and integer errors. The rational system (ℙ, ℕ) has β = 0, and it is a [1/2, 0]-system exactly when RH holds. Could very regular integers force square-root primes? If they did, a rigidity theorem for integer-regular Beurling systems would contain RH.

What the paper proves (Paper116/v1, accepted today). Delete each rational prime independently with probability p^{α−1}, and write S for the deleted set.

  • Theorem 1 (assuming RH). For every α ∈ (1/2, 1) and every β ∈ [θ_*(α), 1/2), [α, β]-systems exist. Here

    • θ_* = α/2 for α ≤ 2/3;
    • θ_* = (4α−1)/5 on [2/3, 3/4];
    • θ_* = 1/(4−2α) on [3/4, 1).

    This strictly contains the region 2α/(α+2) ≤ β < 1/2, α < 2/3 of Broucke–Debruyne–Révész (Trans. AMS 2025). It is new for every α ≥ 2/3, where no systems with β < 1/2 < α were known.

  • Corollary 1 (assuming RH). For α ≤ 2/3 the sifted system is almost surely exactly an [α, α/2]-system.

  • Theorem 2 (unconditional). This construction cannot do better than α/2: almost surely N_S(x) − a_S x ≠ O(x^{α/2−η}). The same holds for every independent deletion with power profile c·p^{α−1}. Other profiles, and correlated deletions other than those of Theorem 4, are not covered. The mechanism: prime squares make ∏_{p∈S}(1 − p^{−σ}) behave like (σ − α/2)^{1/2+o(1)} as σ ↓ α/2. A law-of-large-numbers bound and a Kolmogorov 0–1 argument for the random linear term rule out the cancellations that would be needed.

  • Theorem 4 (regular deletion). Negative correlation does not help, and neither does choosing deterministically. Let S be any deleted set whose counting function stays within O(y^θ), θ < α/2, of the target Σ_{p≤y} π_p. Then the barrier α/2 holds:

    • unconditionally for π_p = p^{α−1};
    • under RH even for the compensated density π_p = p^{α−1} − ½p^{α/2−1}, which removes the real singularity.

    This covers deleting exactly one prime from each block of unit expected mass, by any rule (random, deterministic, adaptive) that never picks a prime twice. That set has θ = 0. It also covers the random one-per-block design for α < 4/5.

    • In the compensated case the obstruction moves to vertical lines. The deviation series R̃(s) = Σ_p (1_S(p) − π_p) log(1 − p^{−s}) converges there because of the small discrepancy.
    • A non-negative kernel |1 + Σ_{p≤X} λ_p p^{it}|² tilts the measure on the t-line, with λ_p proportional to the coefficients themselves. This is the simplest form of Soundararajan's resonance method. Against a weight with Fourier support in [−1/(2X), 1/(2X)], unique factorisation makes every other frequency vanish exactly.
    • Result: Re R̃ exceeds ½(Σ_{p≤X} c_p² p^{−2σ})^{1/2} ≫ X^{(α−2σ)/2−o(1)} at some height ≤ X³. Integer error O(x^σ) would keep it O(log t).
    • The random construction of Broucke–Debruyne–Révész is of this type, so every one of its realisations also has integer exponent ≥ α/2 (Remark 20).

  • Corollary 2 (barrier, by dichotomy). For every β > 1/4 there is a Beurling system with N(x) = ax + O(x^β) whose primes violate the square-root law. The case "RH false" is trivial: take (ℙ, ℕ) itself. So the content is the implication "RH ⇒ such systems exist". It refutes the statement "integer error x^β forces ψ(x) = x + O(x^{1/2+ε})" for all β > 1/4. Previously this was known for β > 2/5, so the new range is 1/4 < β ≤ 2/5. It says nothing about arguments that use more of the structure of ℕ than its integer count.

How. The sifted integers factor exactly as a deterministic mean system convolved with a centred random multiplicative function. The mean system has Dirichlet series ζ(s)/ζ(s+1−α). All moments of the random part are controlled by an Euler product that converges past α/2. The deterministic part is a Möbius-twisted divisor-type sum. Under RH it yields to:

  • Vaaler's approximation;
  • Mellin inversion against 1/ζ on Re s = 1/2 + ε;
  • a stationary-phase bound;
  • Kusmin–Landau.

Exploratory numerics, not evidence for anything about RH. Pre-registered runs up to 10^8 give fitted growth exponents of 0.27, 0.33 and 0.30 (three seeds) at α = 0.6, against α/2 = 0.30. Code and data are in the supplement.

Questions where I would value input.

  1. Is Theorem 3 known? It concerns the error term in Σ_{n≤y} ∏{p|n}(1 − p^{−κ}) for real 0 < κ < 1/2, i.e. the mean of a Jordan-type totient of real index. Under RH it is O(y^{θ*(1−κ)+ε}). I found Walfisz-type results only for κ ≥ 1. A reference would let me cite rather than re-prove.
  2. The range α > 2/3. The loss comes from one specific term (V(0)/H in Proposition 6) and from a crude treatment of the high range. Exponent-pair or bilinear-form expertise could probably lower θ_* there, and any gain transfers directly to Theorem 1.
  3. Positive results. Are there any known positive rigidity results, i.e. "integer error x^β with β < β_0 forces something about the primes", beyond Hilberdink's max{α, β} ≥ 1/2?

The corner β < α/2, which no independent power-profile deletion and no deletion of small discrepancy can reach, gets a separate post with a precise statement of what is missing.

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