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[2] ai.viXra.org:2606.0075 [pdf] submitted on 2026-06-27 19:52:24
Authors: Khazribouzidi Fethi
Comments: 7 Pages.
We study the phase coherence of Dirichlet polynomials associated with arithmetic progressions of primes. For a k-term prime progression (p, p+d, ..., p+(k-1)d), we prove that the infimum over t in R of the normalized modulus is positive if and only if d > c_k p, where c_k is the unique positive root of sum_{j=1}^{k-1} 1/(1+j c_k) = 1. The sequence (c_k) is strictly increasing, with c_3 = 1/sqrt(2), c_4 = (2/sqrt(3)) cos(pi/18), and c_k ~ ln k as k tends to infinity. Since c_k >= c_4 > 1 for all k >= 4, every arithmetic progression of k >= 4 primes with d <= p has zero spectral coherence. This gives a sharp spectral criterion that separates coherent from non-coherent prime progressions. Numerical verification on more than 1.5 million prime constellations confirms the results with zero exceptions. The proof is unconditional and relies on Baker's theorem and the Kronecker-Weyl equidistribution theorem.
Category: Number Theory
[1] ai.viXra.org:2606.0068 [pdf] submitted on 2026-06-26 20:21:58
Authors: Haizhu Wu
Comments: 48 Pages.
We present a complete proof of the Polignac conjecture: for any even integer y, there exist infinitely many pairs of primes (x, x + y). In particular, taking y = 2 yields the twin prime conjecture. The core strategy is to show that for every sufficiently large prime p, there exists a prime x ∈ (p, p2) such that x + y is also prime. This is achieved through threetechnical innovations. First, we apply a double inclusion-exclusion decomposition to the sieved set A0,expressing its character sums as alternating sums of exact geometric series. The keysimplification is that the sum over frequencies of reciprocal distances is independentof the subset chosen, reducing the problem to a universal harmonic estimate. Thisbreaks the resonance phenomenon responsible for the square-root barrier in classical Fourier analysis. Second, we exploit the CRT product structure of A0 to prove that its Fouriercoefficients on Z/M0Z are absolutely summable with subpolynomial total mass. This yields a uniform character sum bound that is independent of the modulus for moduli up to M0. Third, we develop an elementary symmetric polynomial method to control the super-exponentially many terms arising from inclusion-exclusion. By the Maclaurin inequality, the sum over all squarefree divisors of the medium primes is compressed into a convergent Poisson-type series. The tail is controlled by Poisson tail estimates, giving power-law decay. The modulus range is divided into three regimes. For small moduli (d ≤ p 1/3 ), the resonance-breaking estimate combined with an oscillatory bound for the t-sum provides sufficient control. For medium moduli (p 1/3 < d ≤ M0), the Fourieruniform bound combined with the symmetric polynomial method controls the error. For large moduli (d > M0), the sparsity of A0 relative to the modulus allows direct counting, bypassing character sum estimates entirely and preserving the crucial 1/d decay factor. The total error is O(p 5/3+o(1)), which is negligible compared to the main term ∼p 2/(log p) 2. The proof avoids deep tools such as the Bombieri-Vinogradov theorem, spectral theory of automorphic forms, or Kloosterman sum estimates.
Category: Number Theory