Number Theory

2605 Submissions

[2] ai.viXra.org:2605.0009 [pdf] submitted on 2026-05-05 05:23:36

Positivity of L'(x) Implies the Riemann Hypothesis

Authors: Yufei Liu
Comments: 5 Pages.

We prove that if L'(x)=∑_ρ (1/ρ) e^{-x/ρ} > 0 for all x>0, where the sum is taken in the symmetric pairing ρ,1-ρ, then all non-trivial zeros of the Riemann zeta function ζ(s) satisfy Re(ρ)=1/2. The proof uses the arithmetic representation of L'(x) derived in a previous paper, the Guinand--Weil explicit formula, and an asymptotic analysis of the prime sum. Assuming the existence of a zero with real part >1/2 leads to a contradiction by showing that L'(x) would become negative for a suitably chosen sequence of x.
Category: Number Theory

[1] ai.viXra.org:2605.0004 [pdf] submitted on 2026-05-04 01:27:59

Minimum Prime Gaps in Arithmetic Progressions: A Probabilistic Model Based on Hardy--Littlewood Correlations and Empirical Verification on 10^6 Primes

Authors: Giovanni Ferraiuolo
Comments: 6 Pages.

We propose a coherent probabilistic model for consecutive prime gaps inside a fixed arithmetic progression modulo M. The model combines a Cramér-type intensity filtered by the residue class, Hardy—Littlewood two-point correlations via the singular series S(g), and an exponential suppression of intermediate primes. Under natural assumptions, the relative frequencies of small admissible gaps satisfy freq(g2)/freq(g1) ~ S(g2)/S(g1). We test the model on the first 10^6 primes (up to 1.5×10^7) in the digital root classes modulo 9 using exact rational arithmetic in SageMath. The predicted resonance for gap 90 (excess factor 4/3) is observed as +34.5% against +33.3% predicted, an agreement within 1.2 percentage points over 166,567 gaps. For gaps approaching the mean spacing the two-body approximation breaks down, with a sign inversion at gap 198, clearly marking the transition scale. All computations confirm the structural Lemma 1 (zero violations) and the asymptotically stable product freq(g_min) × mean_gap ≈ 2 C2 S(g_min) φ(M).
Category: Number Theory