Density-based structural frameworks for prime numbers, prime gaps, and Euler products

TL;DR

The paper develops a density-based framework for primes, gaps, and zeta zeros, recasting primality, coprimality, and prime pairs as global density objects. It introduces an intrinsic, normalized model for prime gaps and derives structural tensions among Hardy-Littlewood, Cramér, and the Prime Number Theorem, yielding quantitative extreme-gap estimates and a probabilistic interpretation of Goldbach-type representations. In the analytic-number-theory portion, truncated Euler products and a density perspective on ζ(s) generate heuristic criteria for the critical line and zero localization, linking Euler-product oscillations to zero statistics and pair correlations. Overall, the work provides a unifying structural bridge between probabilistic, combinatorial, and analytic approaches to primes, gaps, Goldbach-type problems, and the Riemann zeta function, with testable predictions and a framework for future rigorous development.

Abstract

We develop a unified density-based framework for primality, coprimality, and prime pairs, and introduce an intrinsic normalized model for prime gaps constrained by the Prime Number Theorem. Within this setting, a structural tension between Hardy-Littlewood, Cramer, and PNT predictions emerges, leading to quantitative estimates on the rarity of extreme gaps. Additive representations of even integers are reformulated as local density problems, yielding non-conjectural upper and lower bounds compatible with Hardy-Littlewood heuristics. Finally, the Riemann zeta function is analyzed via truncated Euler products, whose stability and oscillatory structure provide a coherent interpretation of the critical line and prime-based numerical criteria for the localization of non-trivial zeros.

Figures (17)

• Figure 1: $2 \ \bar{p}(n)/n$\ref{['p_n']} (blue) and its heuristic approximation \ref{['p_avg']} (red), for $n\leq 1000$.
• Figure 2: $K_C(n,n)$\ref{['K_C(n,n)_eq']} (blue), $\varphi(n)_{min} \sim 0.208 \ n$ and $\varphi(n)_{max}$\ref{['K_C_bounds']} (red), for $n \leq 1000$ and $w = 11$. As reference, $n/3$, $n/2$ and $2n/3$ (orange).
• Figure 3: Distribution of $K_C(n,n)/n$ values (blue), for $n \leq 200$. As reference, $1-n^{-1}$ (red), $1/3$, $1/2$ and $2/3$ (orange).
• Figure 4: $S_j(n)$\ref{['final_Sj(n)_2']} (blue) compared to $P_{emp}(j,n)$ (red), for $n=10^7$ and $\rho_{max} = 1.92$. As references, the horizontal green lines are $S_j(n) = (\ln n)^{-2}$ and $S_j(n) = n^{-1}$, while the vertical green line is $j = \ln n$.
• Figure 5: $S_j(n)$\ref{['final_Sj(n)_2']} as a function of $j$: $n = 10^{10}$ (black), $n = 10^{25}$ (red), $n = 10^{50}$ (orange), $n = 10^{100}$ (green), $n = 10^{200}$ (blue), $n = 10^{300}$ (violet). Vertical lines correspond to $\rho = 2$ for the considered values of $n$.