False Chebyshev Primes: Prime-Phase Non-Equidistribution and a Weighted Local-Time Law

Abstract

Let ψ be the Chebyshev prime-counting function and π(x) the number of primes up to x. The Chebyshev-prime condition K(p)=li(ψ(p))−li(ψ(p−1))<1 is not equivalent to the sign rule ψ(p)−p>0. We prove that its leading threshold is the midpoint correction ψ(p)−p>12logp, with an explicit higher-order boundary, and define false Chebyshev primes through the resulting transition layer. Our main unconditional theorem shows that ordinary prime counting does not equidistribute logarithmic prime phases: for every nonzero integer combination τ of nontrivial zeta-zero ordinates, π(x)−1∑p≤xpiτ=xiτ/(1+iτ)+o(1). By contrast, the logarithmically uniform weight logp/p restores finite-dimensional Haar equidistribution for rationally independent ordinates. This dichotomy motivates the weighted local-time statistic L+(x)=2∑p≤x,p in the layer p−1/2. Under the Riemann hypothesis and linear independence, we conjecture the one-directional Cesàro law L+(x)/logx→fB(0), where fB(0) is the value at the origin of the limiting density of the symmetrized midpoint field B. Independently, we prove periodic and uniformly transverse quasiperiodic shrinking-target laws via coarea formulas. The remaining obstacles are tangential crossings of the zeta-derived field and transfer to the prime-sampled diagonal window.