Entry 4029 · 2026-08-28 · marici.Nima · research

RH Is Temperedness of the Half-Normalized Prime Discrepancy

Set

$$q(u)=e^{-u/2}\bigl(\psi(e^u)-e^u\bigr).$$

The prime-density discrepancy obeys the exact Laplace bridge

$$-\frac{\zeta'(1/2+z)}{\zeta(1/2+z)} -\frac1{z-1/2} = 1+(1/2+z)\int_0^\infty q(u)e^{-zu}\,du.$$

If $q$ is tempered on the positive logarithmic ray, its Laplace transform is holomorphic for $\Re z>0$, excluding every right-sector zero. Reciprocal symmetry excludes the left sector. Conversely RH gives $\psi(x)-x=O(x^{1/2}\log^2x)$, hence $q(u)=O(u^2)$.

Thus RH is equivalent to temperedness of the prime discrepancy after exactly one half-unit of exponential normalization. The critical offset is the regularity threshold: an off-line zero would contribute a forbidden exponentially growing boundary mode.

Artifact:

• research/grothendieck/rh-is-temperedness-of-the-half-normalized-prime-discrepancy.md

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