The Square-Root Horizon: Squares, Reversal and Prime Clocks in the Distribution of the Primes and the Zeros of the Riemann Zeta Function

Source abstract

We read the primes and the zeros of the Riemann Zeta Function through three exact structures: the residue classes of the divisor r in the rebuild law ∑{r≤x} M(x/r) = 1 for the Mertens function, the squares of the primes, and the reversal of the Euler product. Split by r mod 3 the rebuild law is (1, a(x), −a(x)) for x ≥ 3, with a(x) = ½∑{k≤x}(μ ∗ χ₋₃)(k) carried only by 3 and the primes ≡ 2 (mod 3); split by r mod 4 it is (1, b(x), 0, −b(x)). The zero class reproduces the whole law at x/3, so the classes modulo 3ᵏ form a self-similar tower with an exact energy recursion whose leaves are the values M(x/r); the leaf energy is ≪ x¹⁺ε for every ε > 0 if and only if the Riemann Hypothesis (RH) holds. At every zero ρ of ζ the three classes of the Dirichlet series stand as (0, ½L(ρ, χ₋₃), −½L(ρ, χ₋₃)). An exact sieve to 10¹⁰ in integer and Eisenstein-integer arithmetic gives every class value at 10¹⁰, and the wave amplitudes of a(x)/√x at the first twelve zeros agree with |L(ρ, χ₋₃)/(ρζ′(ρ))| to within 0.0003, including a sixth wave of amplitude 0.0012 against 0.0011 predicted.For consecutive primes p < p′ the primes in [p, pp′) together with p² are exactly the integers there free of the primes below p; with 25 and 49 inserted, the gaps from 5 to 59 are 2, 4, 2, 4, 2, 4, 2, 4, 2, 6, 4, 2, 4, 2, 4, 6 and the classes 6k ± 1 up to 61 stand 9 : 9. Each prime square enters the logarithm of the Euler product with weight ½, and at that weight the lean of the prime count vanishes: over 2¹² ≤ x ≤ 10¹⁰ the mean of (π(x) − li(x))log x/√x is −1.364, that of (J(x) − li(x) + log 2)log x/√x is +0.0013, and the square weight that balances the count, fitted from the data, is 0.4994; the lean of the Liouville sum is carried by its square multiples, L(x) − M(x) = ∑_{d≥2} M(x/d²), and that of the classes 6k ± 1 by the prime squares, which all lie in the class 6k + 1 and at weight ½ reduce it from 1.157 to 0.031 in units of √x/log x.The squares part E(s) = exp(∑ₚ∑_{k≥2} p⁻ᵏˢ/k) of ζ is analytic and zero-free for σ > ½, satisfies E(σ)³|E(σ + it)|⁴|E(σ + 2it)| ≥ 1 there, and has at s = ½ a half-order pole matched by a half-order zero of the prime factor ζ/E, which carries every zero of ζ in σ > ½. In the model with independent uniform phases, the joint limit law of the phases t log p, which unique factorisation makes linearly independent over the rationals, the random prime sum converges almost surely exactly for σ > ½ and its variance is the squares series ∑ₚ p⁻²σ; at heights 5000 to 6000 the distribution of log|ζ(σ + it)| for σ = 0.6, 0.7, 0.8 follows this law, with standard deviations within 2.5%.Reversing the sign of every prime turns ζ(s) into ζ(2s)/ζ(s), the zeros of ζ into poles and the product of the two patterns into ζ(2s); we prove ∑{n≤x} L(x/n) = ⌊√x⌋ and −∑{k≥1} λ(k){x/k} = ⌊√x⌋ for every x > 0, and that if the sawtooths {1/(kt)} approximate the odd-square staircase [⌊t⁻¹ᐟ²⌋ odd] in L²(0,∞) then every zero of ζ in the critical strip lies on the line σ = ½. Computed from exact integrals for up to 2000 sawtooths, the mean of D_N² log N over 1200 ≤ N ≤ 2000 lies 2.5% above ∑_ρ |η(2ρ)|²/|ρ|² = 0.0750, beside 0.6% below 2 + γ_E − log 4π = 0.0462 for d_N² log N with the unit step.Over the 2469 zeros below height 3000 Landau’s formula holds to within 0.0038 at all 35 prime powers up to 100: each prime pulls the zeros toward the reversal p^{iγ} = −1 of its clock, the zeros are 0.65 to 0.68 times as dense near the alignment 2^{iγ} = 1 as on average, and the tenth zero, at 5.4909 turns of the clock of 2, is silenced in the reversed reading by the factor 1 − 2^{1−2s}. In this language, the hypothesis (RH) is the statement that every pole of the reversed pattern lies on the vertical line through its centre s = ½.We state the axiom of harmonic independence: the prime waves, whose frequencies log p are linearly independent by unique factorisation, are related only through the mechanism of their construction, and a zero off the critical line would be a relation without a mechanism.