The global Rosser--Schoenfeld reciprocal-prime lower coefficient is below 1/306

Statement

The claim concerns the lower side of the reciprocal-prime Mertens error on its full natural range. The coefficient is obtained by evaluating a decreasing tail coefficient at the splice point $10^8$.

Why a splice is available

Rosser and Schoenfeld proved that $A_1(x)$ is positive for $1<x\leq10^8$. Any negative lower comparison of the stated form is therefore automatic on that entire interval.

For larger $x$, Axler's estimate supplies a lower bound with the stronger decay terms $\log^{-3}x$ and $\log^{-4}x$.

Tail normalization

Write $L=\log x$ and factor the tail estimate into the desired $L^{-2}$ shape. The remaining coefficient function is strictly decreasing for $L>0$, so its largest value after the splice occurs at $L=\log(10^8)$.

For $x>10^8$ the inequality is strict because the coefficient function has already decreased below its endpoint value.

1. Use positivity on $1<x\leq10^8$.
2. Normalize Axler's tail estimate by $\log^2x$.
3. Apply monotonicity of $g$ beyond the splice.

Certificate role

The pinned certificate evaluates $\log(10^8)$ and $C_0$ with directed 256-bit Arb arithmetic. It proves the strict rational comparison $C_0<1/306$, confirms that $1/307<C_0$, and verifies an improvement factor greater than $153$ relative to the earlier coefficient $1/2$.

The certificate does not encode the cited positivity theorem, Axler's analytic estimate, or the symbolic monotonicity argument. Those are the written proof obligations that connect the certified constants to the global theorem.