Pith Number

pith:L7R2BJIF

pith:2026:L7R2BJIFR7KJMVIZGXD2322UW7

not attested not anchored not stored refs pending

Paratrophic Determinants over $\mathbb{Z}/N\mathbb{Z}$ via Discrete Fourier Transform

Hang Liu

Paratrophic determinants attached to the multiplicative semigroup Z/NZ factor into products of group determinants indexed by divisors of N, using discrete Fourier, cosine, and sine transforms.

arxiv:2603.14795 v3 · 2026-03-16 · math.NT

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\pithnumber{L7R2BJIFR7KJMVIZGXD2322UW7}

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Record completeness

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4 Citations open

5 Replications open

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Claims

C1strongest claim

via discrete Fourier, cosine, and sine transforms, these determinants factor into products of group determinants indexed by d|N

C2weakest assumption

That the paratrophic determinants are defined in a manner that permits direct application of the discrete Fourier, cosine, and sine transforms to achieve the stated factorization over the semigroup Z/NZ.

C3one line summary

Paratrophic determinants over Z/NZ factor via discrete Fourier transforms into products of group determinants for d dividing N, giving explicit formulas for Bernoulli and tangent cases and correcting a conjecture.

Receipt and verification

First computed	2026-06-09T01:05:16.047924Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

5fe3a0a5058fd496551935c7adeb54b7f08c0a48b3448511c28fc14cdbc691e2

Aliases

arxiv: 2603.14795 · arxiv_version: 2603.14795v3 · doi: 10.48550/arxiv.2603.14795 · pith_short_12: L7R2BJIFR7KJ · pith_short_16: L7R2BJIFR7KJMVIZ · pith_short_8: L7R2BJIF

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/L7R2BJIFR7KJMVIZGXD2322UW7 \
  | 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: 5fe3a0a5058fd496551935c7adeb54b7f08c0a48b3448511c28fc14cdbc691e2

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "6c867cf1577e3d34566f89410e25e889fcdb18f7c2472924209a325a8b5eb598",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-03-16T03:51:29Z",
    "title_canon_sha256": "e15ceb1f8dba856fdbb47f619faa5b3e1c5365b63f9a355b7b7d01f1dff23245"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2603.14795",
    "kind": "arxiv",
    "version": 3
  }
}