# Axler compact prime-counting denominator search

Date: 2026-07-19

## Named published constant

Christian Axler, *Effective Estimates for Some Functions Defined over
Primes*, Integers **24** (2024), paper A34, Proposition 4, proves

\[
\pi(x)<
\frac{x}{
\log x-1-\frac1{\log x}-\frac3{\log^2x}
-\frac{70.935}{\log^3x}
}
\qquad(x\geq29.53).
\]

The arXiv version `2203.05917v4` labels the same statement
Proposition 4.5. The published range is the plain inequality
\(x\geq29.53\), not \(\log x\geq29.53\).

The target preserves the entire denominator shape and range while
replacing \(70.935\) by \(44.053\).

## Primary inputs

- Axler, Integers **24** (2024), Proposition 4.
- Jan Buethe, *An Analytic Method for Bounding \(\psi(x)\)*,
  Mathematics of Computation **87** (2018), 1991--2009.
- Andrew Fiori, Habiba Kadiri, and Joshua Swidinsky, *Sharper Bounds
  for the Error Term in the Prime Number Theorem*, Research in Number
  Theory **9** (2023), article 63; arXiv `2206.12557v2`.
- C-0051, the exact same-form sharpening of the FKS global
  \(\pi-\operatorname{Li}\) amplitude from \(9.2211\) to
  \(9.221056\).

## Prior-art search

Primary-source and exact-string searches through 2026-07-19 included:

- `"44.053" "prime counting function"`;
- `"44.053" "70.935" prime counting`;
- `"44.052426" prime counting`;
- `"70.935" "prime counting function"`;
- `"29.53" "70.935" pi(x)`;
- `site:arxiv.org "44.053" "pi(x)"`;
- the complete target denominator formula;
- Axler's paper and forward references to Fiori--Kadiri--Swidinsky
  explicit prime-counting bounds.

The searches recovered Axler's \(70.935\) statement and the FKS error
bounds but no same-range result with coefficient \(44.053\) or
smaller.

## Exact optimization

For \(L\geq40\), a rational majorant for
\(Le^{-L}\operatorname{Ei}(L)\) and all 164 FKS Table 4 rows reduce
the admissible coefficient to 163 exact rational inequalities. Their
unique maximum occurs on \([2100,2200]\):

\[
\frac{
2709976113860085547103395243942841453423728245400
}{
61517068085059134496927148872832330483236342381
}
=44.05242639511077\ldots.
\]

Thus the short decimal \(44.053\) is admissible. It is not claimed to
be the least coefficient for the actual prime-counting function; it
is an upward rounding of the exact published-envelope requirement.

## Consequence

The coefficient decreases by

\[
70.935-44.053=26.882.
\]

Both denominators are positive on \(x\geq29.53\), so the new
right-hand side is strictly smaller at every point of the published
range. This is a direct strengthening of a compact bound intended for
reuse in estimates for \(1/\pi(x)\), the \(n\)-th prime, and related
explicit inequalities.

## Source hashes

- Axler journal PDF:
  `c093bd507a7e130d1c28ebe8a8707d811a39f8c60ba6cc598afc4bc6e2e522f9`.
- Axler journal text:
  `5597897b3fed188cbaee881fb8a0048ecfdcf265edbf2dc1fb69a8787781d50e`.
- Buethe PDF:
  `350f96313810ded3e8eaf2af6006b227c8834235e177f49eabb3d43d4931c4d1`.
- Fiori--Kadiri--Swidinsky PDF:
  `405e5dcbca6067fa436141b8a08c79b81bc12deca03bfe12d667428c36b50fbb`.
- Fiori--Kadiri--Swidinsky text:
  `9c926ea1ed333e8008c245f8f5bdd87c72fff0eb1647f87d7e02fbf198169edf`.

## Decision

Proceed. The named coefficient and identical range are primary-source
verified, no prior same-form \(44.053\) result was found, and every
range handoff has a positive exact or directed-rounding margin.
