Number Theory

Positivity of L'(x) Implies the Riemann Hypothesis

Authors: Yufei Liu

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.

Comments: 5 Pages.

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[v1] 2026-05-05 05:23:36

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