Pith Number

pith:X5UYGB2A

pith:2026:X5UYGB2A72CWGIVTXBJ7BA4CXN

not attested not anchored not stored refs pending

Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

Runbo Li

Variants of Harman's sieve produce mean value theorems for primes in arithmetic progressions to moduli as large as x to the 9/17 in bilinear form.

arxiv:2602.20917 v6 · 2026-02-24 · math.NT

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\usepackage{pith}
\pithnumber{X5UYGB2A72CWGIVTXBJ7BA4CXN}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications 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 obtain some mean value theorems for primes with bilinear forms of moduli up to x^{9/17} or with trilinear forms of moduli up to x^{17/32}. As a by-product, we obtain new upper and lower bounds for π(x; q, a) that hold for almost all moduli q.

C2weakest assumption

The arithmetic information from the cited works of many authors combines with the variants of Harman's sieve to produce majorants and minorants that satisfy the required Bombieri-Vinogradov type mean value theorems without further restrictions.

C3one line summary

Refinements of Harman's sieve produce Bombieri-Vinogradov mean value theorems for primes in APs with bilinear moduli up to x^{9/17} and trilinear up to x^{17/32}, yielding new upper and lower bounds for π(x; q, a) for almost all q.

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-28T02:04:46.485827Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

bf69830740fe856322b3b853f08382bb7646b56088c7d88026395268fadc6484

Aliases

arxiv: 2602.20917 · arxiv_version: 2602.20917v6 · doi: 10.48550/arxiv.2602.20917 · pith_short_12: X5UYGB2A72CW · pith_short_16: X5UYGB2A72CWGIVT · pith_short_8: X5UYGB2A

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/X5UYGB2A72CWGIVTXBJ7BA4CXN \
  | 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: bf69830740fe856322b3b853f08382bb7646b56088c7d88026395268fadc6484

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6a3e5a9112604fed3e3902a4fd723f686aba4696945526ba754321b402939707",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-02-24T13:54:22Z",
    "title_canon_sha256": "cd05b32b2d2f39c43ae24b441398c23ab591e4ca66b4dda962602dfe95b034c9"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2602.20917",
    "kind": "arxiv",
    "version": 6
  }
}