PRH | Aux | 4.7 • Prime Number Theorem: Near-One Blur

This article gives a Near-One Blur formulation of the classical logarithmic zero-free region for the Riemann zeta-function and then reads the Prime Number Theorem through a second, cumulative blur. The proof is deliberately arranged as a dependency-audited argument. Before the zero-free region is proved, the only prime-side input is the Euler product for

$$
-\frac{\zeta^{\prime}}{\zeta}(s)
$$

in the half-plane $\Re s>1$. The remaining pre-PNT analytic inputs are standard zeta-function facts obtained from analytic continuation, the functional equation, the Hadamard product, and the classical local zero-counting estimate. No asymptotic statement about $\psi(x), \vartheta(x)$, or $\pi(x)$ is used in the proof of the zero-free region.

This is a heat-kernel version of the classical de la Vallée Poussin positivity mechanism. Instead of a fixed finite trigonometric polynomial, the proof uses a guarded positive heat channel. The heat kernel on the phase circle opens a channel from the Euler product of $-\zeta^{\prime} / \zeta$ to a nonnegative prime-side observable. Local meromorphic guards then allow the same observable to be read as a pole-zero balance near $s=1$.

The main technical step is a Guarded Heat-Blur Exclusion Lemma. It says that whenever a positive channel sees the pole at $s=1$ with coefficient 1 , sees a possible zero at height $T$ with coefficient $A>1$, and loses only $O(\log (2+|T|))$ information in the remaining harmonics, then no zero may lie within distance $c / \log (2+|T|)$ of the line $\Re s=1$. For the zeta-function the heat coefficient is

$$
A_\tau=2 e^{-\tau}>1 \quad(0<\tau<\log 2),
$$

so the lemma gives the usual logarithmic zero-free region: $\zeta$ has no zeros, apart from the pole point $s=1$ which is not a zero, in

$$
\Re s \geq 1-\frac{c}{\log (2+|\Im s|)} .
$$

Only after this zero-free region has been obtained do we pass to primes. A Gaussian cumulative Perron blur gives a smoothed Chebyshev law, and a positive monotone deblurring lemma, using only the elementary one-sided Chebyshev bound $\psi(x) \ll x$, recovers $\psi(x) \sim x$ and hence $\pi(x) \sim x / \log x$.