Pith Number

pith:K4H3EVX4

pith:2026:K4H3EVX4VW3AUYAHAKASSGJFPK

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The $S$-$E$ route to the Chebyshev bounds for the prime-counting function

Kai Hubbard

An order-of-magnitude bound on the weighted prime sum S(x) implies the Chebyshev bounds for the prime-counting function.

arxiv:2604.21946 v2 · 2026-04-22 · math.GM

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\pithnumber{K4H3EVX4VW3AUYAHAKASSGJFPK}

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4 Citations open

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The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We prove that the order-of-magnitude estimate S(x) ≍ sqrt(x / log x) implies the Chebyshev bounds π(x) ≍ x / log x through a short and transparent chain of inequalities. The mechanism passes through E(x), which we show satisfies E(x) ≍ π(x) whenever the size estimate for S(x) holds.

C2weakest assumption

That the specific inequalities relating E(x) to π(x) hold with the claimed constants and without hidden restrictions when S(x) satisfies the given order-of-magnitude bound.

C3one line summary

S(x) ≍ sqrt(x / log x) implies π(x) ≍ x / log x because E(x) ≍ π(x) under that assumption, with the S estimate following from Mertens' theorem.

Receipt and verification

First computed	2026-06-02T02:04:53.409875Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

570fb256fcadb60a600702812919257a8f6119e8bc5061455c30bb983fbae380

Aliases

arxiv: 2604.21946 · arxiv_version: 2604.21946v2 · doi: 10.48550/arxiv.2604.21946 · pith_short_12: K4H3EVX4VW3A · pith_short_16: K4H3EVX4VW3AUYAH · pith_short_8: K4H3EVX4

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/K4H3EVX4VW3AUYAHAKASSGJFPK \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 570fb256fcadb60a600702812919257a8f6119e8bc5061455c30bb983fbae380

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "a22136441ed90d78d06e0f6a49fda2b73357ec8443445f4ab3629d80b12d3949",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.GM",
    "submitted_at": "2026-04-22T05:56:29Z",
    "title_canon_sha256": "95e2fb6dd7b387d52691412831bab0ee8fc58d2a9e5e202edc1b6411b1e4446b"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.21946",
    "kind": "arxiv",
    "version": 2
  }
}