Gaussian Diagonal Dominance for Explicit Formulas in Analytic Number Theory

We develop a general technique for establishing diagonal dominance of Weil-type explicit formulas when evaluated on Gaussian test functions centered at prime logarithms. The sole arithmetic input is the discreteness of integers: the Fundamental Theorem of Arithmetic ensures that distinct prime powers have distinct logarithms, providing a quantitative gap. The sole analytic input is Gaussian suppression of nonzero phase mismatches. The main result gives an explicit bound: for the off-diagonal contribution O_k(η) to the prime-sum side of any explicit formula evaluated on h_k(x) = Σ a_j ψ_η(x − log p_j), we prove |O_k(η)| ≤ Cη Σ|a_j| log p_j + C exp(−c/η²) Σ|a_j| log p_j, where C and c are absolute constants independent of k, η, and the weights a_j. The proof uses a multiplicative shell decomposition: prime powers n ≠ p_j are grouped by their multiplicative distance from p_j into dyadic shells, with each shell contributing a bounded amount controlled by the Gaussian width. The bound is elementary (uses only FTA and calculus, no transcendence theory), explicit (all constants are computable and given in the appendix: C₃ ≈ 5.01, c ≈ 0.240), universal (applies to any L-function with Euler product: Riemann zeta, Dirichlet L-functions, automorphic L-functions), and asymptotically sharp (the off-diagonal/diagonal ratio is O(η) → 0). A practical analysis identifies the five dominant off-diagonal pairs contributing ~80% of |O_k|: the pairs (8,7), (9,11), (16,13), (25,23), (32,31) — all cases where a small prime power is close to a prime. These pairs are fixed and their contribution is explicitly computable. The technique reduces positivity questions for explicit formulas to two independent problems that no longer involve the arithmetic of primes: (1) control of the archimedean term involving Γ'/Γ (specific to each L-function), and (2) density of the Gaussian test family in the space of admissible test functions (a problem in approximation theory). The primes have already done their work in creating the positive diagonal. Applications are discussed for the Riemann zeta function (Weil criterion), Dirichlet L-functions (GRH), and general L-functions with Euler products. For the Weil application with spectral weights a_j = (log p_j)²/√p_j, the off-diagonal/diagonal ratio is O(1/log k), giving diagonal dominance for all sufficiently large k. Numerical verification at k = 20 confirms the bound is conservative by a factor ~5. Limitations are stated explicitly: the technique controls only the prime-sum side of explicit formulas; archimedean control and density remain as separate open problems.