D_χ(x)
Jump Positions
J_χ Running
C_χ(N) Product
Compare χ
Jump Table
Theory
Dχ(x) — smooth except at x=p−1 where χ(p)≠0. Each discontinuity = a prime in the support of χ.
—
Jump size χ(p)/p(p−1) at each prime. Teal = positive (χ(p)=+1), coral = negative (χ(p)=−1), gray = zero (χ(p)=0).
—
Running sum J_χ(N) = Σ_{p≤N} χ(p)/p(p−1). Converges to J_χ as N→∞.
—
Euler-style product C_χ(N) = Π_{p≤N}(1 − χ(p)/(p²−1)). For principal χ→C≈0.5307; for Liouville λ: C_λ=Π(p²/(p²−1))=ζ(2)=π²/6≈1.6449.
—
J_χ(N) running sum for all characters simultaneously. Shows how each character "tunes" the detector.
—
| Prime p | p mod q | χ(p) | Jump χ(p)/p(p−1) | J_χ running | C_χ factor |
|---|---|---|---|---|---|
| Click ▶ Compute to populate | |||||
The Character-Weighted Jump Theorem
Theorem — D_χ Jump Formula (Wessen Getachew)
Let χ be a Dirichlet character. Define
Dχ(x) = −Σp > x+1, χ(p)≠0 χ(p) / (p² − x − 1)
Then Dχ(x) is smooth on ℝ \ {p−1 : χ(p)≠0}, and at each such p−1,
Dχ(x⁺) − Dχ(x⁻) = χ(p) / p(p−1)
Composites and primes with χ(p)=0 cause no jump. For χ=χ₁ (principal), this reduces exactly to the original Jump Theorem.
Dχ(x) = −Σp > x+1, χ(p)≠0 χ(p) / (p² − x − 1)
Then Dχ(x) is smooth on ℝ \ {p−1 : χ(p)≠0}, and at each such p−1,
Dχ(x⁺) − Dχ(x⁻) = χ(p) / p(p−1)
Composites and primes with χ(p)=0 cause no jump. For χ=χ₁ (principal), this reduces exactly to the original Jump Theorem.
Theorem — J_χ Series Identity
Jχ = Σp χ(p)/p(p−1) = Σk≥2 P(k,χ)
where P(k,χ) = Σp χ(p)/pk is the prime zeta function twisted by χ.
Proof: expand 1/(p(p−1)) = Σk≥2 1/pk and swap sums (justified by absolute convergence since |χ(p)|≤1).
where P(k,χ) = Σp χ(p)/pk is the prime zeta function twisted by χ.
Proof: expand 1/(p(p−1)) = Σk≥2 1/pk and swap sums (justified by absolute convergence since |χ(p)|≤1).
Connection to Dirichlet L-functions. The constant Jχ is related to the logarithmic derivative L'/L(s,χ) at s=2. While not identical to L'(2,χ)/L(2,χ), it is a discrete sum over primes that appears naturally in explicit formulae for the character-weighted error term Eχ(N). This opens the door to studying the Generalised Riemann Hypothesis via the geometric lift framework.
The C_χ Euler Product
Definition — Generalised Lift Constant
Cχ(N) = Πp≤N (1 − χ(p)/(p²−1))
For χ=χ₁: Cχ₁(∞) = C = Πp(p²−2)/(p²−1) ≈ 0.530711806246
For non-principal χ: Cχ measures the character-weighted density of lift-surviving residues. Jχ = −(d/ds log Cχ(s))|s→∞ in a suitable sense.
For χ=χ₁: Cχ₁(∞) = C = Πp(p²−2)/(p²−1) ≈ 0.530711806246
For non-principal χ: Cχ measures the character-weighted density of lift-surviving residues. Jχ = −(d/ds log Cχ(s))|s→∞ in a suitable sense.
The "Quantum Spin" Interpretation. The principal character Dχ₁(x) acts as a classical gravity detector — it reacts to every prime with equal weight. The character-weighted Dχ(x) is a quantum spin detector: by tuning χ, it reads not just the prime's position but its arithmetic congruence class. For χ₄ (mod 4), positive jumps at p≡1(4) and negative jumps at p≡3(4) make the detector "see" the spin of the prime — split vs. inert in ℤ[i].
Liouville character λ. Since λ(p)=−1 for all primes, C_λ(N) = Π_{p≤N}(1+1/(p²−1)) = Π_{p≤N} p²/(p²−1) → ζ(2) = π²/6 ≈ 1.6449 as N→∞. This connects the character-weighted lift framework directly to the Basel problem. Meanwhile J_λ = −J ≈ −0.7732, the exact negative of the principal constant.
Parity and GRH
Parity obstruction. For the non-principal character mod 4, jumps alternate in sign. This cancellation can overcome the parity obstruction for certain prime subsets — because χ distinguishes numbers with even vs. odd prime factor count in a given residue class. The error term Eχ(N) = Rχ(N) − Cχ would then be related to sums over zeros of L(s,χ), connecting the lift framework directly to GRH.
Novelty statement. While the individual components (Dirichlet characters, prime zeta functions, L-functions) are classical, the specific structure Dχ(x) as a dynamic diagnostic tool for prime congruence classes, and the identity Jχ = Σ_{k≥2} P(k,χ) as a decomposition of the lift constant, appear to be original to this framework.