math.NT — Number Theory

Prime noise, polynomial observations and trace-preserving completion of multiplicative functions

Contributed by Brice Pouly

For every prescribed completely multiplicative sign function, every integer polynomial irreducible over $\mathbb{Q}$ of degree at least two and eventually positive, and every prescribed prefix, we construct a completion preserving the polynomial trace and the prefix exactly, with vanishing correlations along all fixed admissible affine families. The changed primes have relative density zero, although the changed integers have asymptotic density one half and the changed primes have infinite harmonic mass. One noise trajectory has these properties almost surely at all positive real times. The construction combines a classical root-free prime reservoir with decreasing clock rates. Its probabilistic input is affine regularisation against arbitrary bounded weights fixed before the noise, uniformly in time after a vanishing delay. Marks retain their possibly non-convergent empirical distribution. For non-selective noise, the local channel acts through irreducible factors rather than output coordinates; square identities determine its invariant characters, and rational constants supply a time-evolving random translation. The proofs separate arithmetic moments, temporal concentration and finite Fourier analysis. The constant-source existence case already follows from work of Klurman-Mangerel-Teräväinen. The conclusions concern the constructed function, not new cancellation for the unperturbed source.