Short-Interval Universality Transfer: From Hurwitz and Beurling to the Prime Zeta Function

Start with the O(H) short-interval mean-square bound for ζ(s,α) and the probabilistic-approximation framework for Beurling ζ_P(s+iτ). Transfer these techniques to series closely tied to log ζ, then to P(s)=∑ μ(k)/k·log ζ(ks), aiming for an effectivized universality statement for P(s+iτ) in windows [T−H,T+H] when H is a small power of T. Use this to derive nontrivial zero-density bounds and to quantify oscillations in P(s) akin to Révész’s oscillation program for Beurling primes. Cross-check with the zero-free region and zero-distribution results for P(s). This differs from prior work by pushing short-interval and probabilistic approximation tools into the prime zeta setting via its log-ζ representation, clarifying to what extent P(s) inherits universality-like behavior from ζ and where the Möbius-weighted scaling breaks it. Even partial short-interval universality for P(s) would be a qualitative leap, enabling approximation and zero-density tools tailored to P(s), and reconciling conflicting observations about degeneracy tendencies versus local variability. The impact is a transferable toolkit for effectivizing universality across non-Euler zetas and derived Dirichlet series, with concrete zero-density and oscillation corollaries deepening the emerging theory of P(s) and sharpening numerical expectations for its zero landscape.

References:

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