---
id: C-0070
type: claim
title: "Johnston--Yang's global Vinogradov--Korobov psi decay improves to 0.2043"
status: proved-unconditional
family: adjacent-unconditional
statement: >
  For every real x>=23, |psi(x)-x| is less than
  0.026*x*(log x)^1.801 times exp(-0.2043*(log x)^(3/5)/
  (log log x)^(1/5)). This preserves Johnston--Yang's amplitude,
  logarithmic power, and range while replacing their decay 0.1853.
verifier: canon/witnesses/C-0070/verify.py
depends: [C-0067]
supersedes: []
sources:
  - literature/2026-07-19-johnston-yang-vk-global-psi-decay-02043-search.md
  - literature/2026-07-19-johnston-yang-vk-global-psi-decay-02043-transcript.md
  - experiments/johnston_yang_vk_global_psi_decay_02043/verify.py
  - canon/witnesses/C-0070/PROOF.md
  - canon/witnesses/C-0070/PIN.md
  - scratch/adjacent-unconditional--johnston-yang-vk-global-psi-decay-improvement/INDEPENDENT-REVIEW.md
  - scratch/adjacent-unconditional--johnston-yang-vk-global-psi-decay-improvement/RESULT.md
tags: [chebyshev-psi, explicit-bound, pnt-error, vinogradov-korobov, zero-free-region, johnston-yang, yang-thesis, interval-arithmetic]
caveats: >
  The proof interiorizes Yang's open denominator 51.34 to 51.3401.
  Johnston--Yang's published zero-density constants are pinned
  primary-source inputs rather than independently rederived theorems.
  The decimal 0.2043 is not claimed optimal, and no novelty is claimed
  for the asymptotic mechanism or its theoretical limiting coefficient.
---

## Statement

Let

\[
L=\log x,\qquad
r(L)=\frac{L^{3/5}}{(\log L)^{1/5}}.
\]

For every real \(x\geq23\),

\[
\boxed{
|\psi(x)-x|
<
0.026\,xL^{1.801}\exp(-0.2043r(L)).
}
\tag{1}
\]

Johnston--Yang, JMAA **527** (2023), article 127460, Theorem 1.4,
prove (1) with decay \(0.1853\). The amplitude, logarithmic power, and
range are unchanged. The exact increase is

\[
0.2043-0.1853
=0.019
=\frac{19}{1000},
\tag{2}
\]

or \(10.2536\ldots\%\). Thus (1) is strictly smaller than the
published envelope at every \(x\geq23\).

## Mechanism

Yang's 2025 thesis, Theorem 1.3, proves the open global region

\[
\zeta(\sigma+it)\neq0
\quad(t\geq3),\qquad
\sigma>
1-\frac{1}{
51.34(\log t)^{2/3}(\log\log t)^{1/3}}.
\tag{3}
\]

Set \(c_*=51.3401\). Its closed boundary lies strictly inside (3).
For

\[
\nu_c(t)=
\frac1{c(\log t)^{2/3}(\log\log t)^{1/3}},
\qquad
T=\min_{t\geq H}t x^{\nu_c(t)},
\]

where \(H=3000175332800\), direct minimization gives

\[
\log T\geq Q(c)r(L),
\]

\[
Q(c)=
\left(\frac5{3c^3}\right)^{1/5}
\left[
\left(\frac32\right)^{2/5}
+
\left(\frac23\right)^{3/5}
\right].
\tag{4}
\]

At \(c=c_*\),

\[
Q(c_*)=
0.204352980832328860089304386414397\ldots.
\tag{5}
\]

The proof replaces two defects in Johnston--Yang's Appendix A
calculation:

1. their printed \(B_2=0.18525\) with
   \(\sigma=0.9999932\) yields only
   \(0.1852466408<0.1853\);
2. their displayed \(s_3''\) denominator \(B_2r\) is a typo, since
   \(T\geq e^{B_2r}\) gives \(4.3128L^{0.6}e^{-B_2r}\).

The source integration transcription is also repaired. Rather than
locating its maximizing height, use the definition of \(T\) directly
and take zero-free cutoff values \(U\downarrow T\). This includes a
possible zero at height \(T\) and gives the valid limiting bound
\(2N(\sigma,T)/T\).

Use

\[
L_0=5\cdot10^8,\qquad
\sigma=0.99996,\qquad
B_3=0.21715,
\]

\[
C_1=17.418,\qquad C_2=2.077.
\tag{6}
\]

Johnston--Yang's directed interpolation between the
\(\sigma=0.999\) and \(1\) rows justifies the two density constants.
A fixed trial \(\theta=0.082\) proves

\[
Q(c_*)r(L)\leq\log T\leq B_3r(L)
\quad(L\geq L_0).
\tag{7}
\]

Substitution into their Section 5 argument gives

\[
\begin{aligned}
s_1={}&
\frac{e^{-L/2}\log^2(H/2\pi)}{2\pi}\\
&+e^{-(1-\sigma)L}
\left(
\frac{B_3^2r^2}{2\pi}
-\frac{\log^2(H/2\pi)}{2\pi}
+1.8642
\right),\\
s_2={}&
2C_1e^{aQr}(Qr)^p
+2C_2e^{-Qr}(Qr)^2,\\
s_3={}&
4.3128L^{0.6}e^{-Qr},
\end{aligned}
\tag{8}
\]

where \(p=5-2\sigma\) and \(a=(5-8\sigma)/3<0\). The native first
density decay is

\[
-aQ
=0.204331183181040078344228193946513\ldots
>0.2043,
\tag{9}
\]

and \((3/5)p=1.800048<1.801\). The verifier proves every term in
\((s_1+s_2+s_3)/(0.026L^{1.801}e^{-0.2043r})\) decreases for
\(L\geq L_0\). At the endpoint,

\[
\frac{s_1+s_2+s_3}{
0.026L_0^{1.801}e^{-0.2043r(L_0)}}
=
0.1087515463339506526\ldots<1.
\tag{10}
\]

The remaining ranges are closed as follows:

- \(23\leq x\leq59\): exact prime-power intervals, with minimum
  margin \(0.12706925145\ldots\);
- \(59<x\leq e^{58}\): Johnston--Yang's Büthe bound, with endpoint
  log-margin \(0.89003947\ldots\);
- \(e^{58}\leq x\leq e^{L_0}\): C-0067, with positive endpoint
  log-margins \(0.30211776\ldots\) and \(1125.8910\ldots\).

The C-0067 comparison is unimodal with no interior minimum. These
splices and (10) prove (1).

```text
exact_map: Yang open global VK denominator 51.34 -> interiorized closed input c*=51.3401 -> exact minimization Q(c*) -> repaired Johnston--Yang density and explicit-formula tail -> finite/Büthe/C-0067 splice -> same 0.026 amplitude, 1.801 power, and x>=23 range with decay 0.2043
counterfeit: substitute 51.34 into the old Appendix A decimals, retain the insufficient printed B2=0.18525, or use the printed s3 denominator B2r
counterfeit_failure: PASS -- the verifier exposes the printed decay defect, derives Q(c) analytically, repairs the zero-count endpoint and s3 term, and certifies every range and derivative reduction
success_delta: PASS -- the named global VK decay rises by 0.019 (10.2536%) with unchanged amplitude, power, and range; the asymptotically superior VK-shaped theorem clears C-0067's consequence bar
failure_delta: PASS -- any source mismatch, nonpositive finite/Büthe/C-0067/tail margin, explicit-formula range failure, or derivative failure leaves Johnston--Yang's 0.1853 unchanged
stop_condition: prior same-form all-x>=23 bound with decay >=0.2043, source or range mismatch, failure to reproduce the repaired c=57.54 regime, nonpositive directed margin, or failure to preserve 0.026 and 1.801
```

## Witness

The 256-bit Arb verifier:

1. hash-pins Johnston--Yang v2, Yang's thesis, C-0067, and a 2026
   forward use of the \(0.1853\) constant;
2. checks theorem, density-interpolation, and table-row source anchors;
3. certifies the open-to-closed denominator padding;
4. proves the exact \(Q(c)\) formula and the upper-\(T\) trial bound;
5. encodes every finite, bridge, explicit-formula, and termwise
   monotonicity reduction;
6. proves the endpoint ratio (10) with margin greater than \(0.891\);
7. exposes the printed \(B_2\) defect; and
8. reruns the full repaired \(c=57.54\) tail, including its range and
   derivative conditions.

The independent checker uses 80-digit `mpmath`, separate formulas,
geometric sampling, and a numerical stationary-point solve. It is
corroborative rather than the directed proof. A read-only hostile review
initially found four coverage defects; after repair, the same reviewer
returned closure PASS with no remaining finding.

## What survives

The value \(0.2043\) is not optimal. Formula (5) is only
\(5.3\cdot10^{-5}\) above the stated decimal, and the density parameter
\(\sigma\), handoff \(L_0\), or a stronger global VK denominator can
change the best certifiable value.

C-0067 remains sharper over a substantial finite range, while (1) has
the asymptotically superior Vinogradov--Korobov shape. Neither record
supersedes the other pointwise. The reusable output is the repaired,
parameterized Johnston--Yang tail certificate.

## Provenance

Primary sources:

- Daniel R. Johnston and Andrew Yang, *Some explicit estimates for the
  error term in the prime number theorem*, JMAA **527** (2023),
  article 127460, arXiv:2204.01980v2.
- Andrew Yang, *Explicit zero-free regions for the Riemann
  zeta-function*, UNSW PhD thesis (2025), DOI
  `10.26190/unsworks/31825`, Theorem 1.3.
- C-0067 for the low and middle global absolute-\(\psi\) splice.

A targeted exact-query and forward-use search through 2026-07-19 found
no same-amplitude, same-power, same-range numerical theorem with decay
larger than \(0.1853\). Later Johnston and Bellotti results establish
asymptotic transfers rather than the same finite explicit envelope. The
proof, directed certificate, independent reconstruction, and hostile
closure review were completed on 2026-07-19.
