Pith Number

pith:CED2MSZN

pith:2026:CED2MSZNE744WAPGM26VEYJ4H2

not attested not anchored not stored refs pending

Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions

XinHang Ji

For large prime P, the sum over characters mod P of low-lying zeros of L(s, chi) on the critical line in intervals of length T is at least order T squared times P times sqrt(log P), even for T as small as 1 over sqrt(log P).

arxiv:2605.09282 v2 · 2026-05-10 · math.NT

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\usepackage{pith}
\pithnumber{CED2MSZNE744WAPGM26VEYJ4H2}

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Record completeness

1 Bitcoin timestamp

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

for a sufficiently large prime P and real number T in [a1/sqrt(log P), 1], we prove that sum_{chi mod P} N0(T, chi) >> T^2 P sqrt(log P)

C2weakest assumption

The high-dimensional Mellin transform framework can be applied to the multi-variable series from the mollifier without residual cross-terms or error terms that would invalidate the lower bound extraction in the stated short-interval range.

C3one line summary

For large prime P and T at least on the order of 1 over sqrt(log P), the summed count of low-lying zeros on the critical line over characters mod P satisfies sum N0(T, chi) much greater than T squared P sqrt(log P).

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-20T00:04:35.149185Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

1107a64b2d27f9cb01e666bd52613c3e9826f05cbe09a488ddebee5f0dd61e55

Aliases

arxiv: 2605.09282 · arxiv_version: 2605.09282v2 · doi: 10.48550/arxiv.2605.09282 · pith_short_12: CED2MSZNE744 · pith_short_16: CED2MSZNE744WAPG · pith_short_8: CED2MSZN

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/CED2MSZNE744WAPGM26VEYJ4H2 \
  | 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: 1107a64b2d27f9cb01e666bd52613c3e9826f05cbe09a488ddebee5f0dd61e55

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "e88768dea9de8aea70fc669ca92abdc21635aa446358180245e8caa667cccaa7",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-10T03:14:41Z",
    "title_canon_sha256": "69e8e60aa9a10cef25066b623441847d43f6b8975d9ae522812dd1aecae37f10"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.09282",
    "kind": "arxiv",
    "version": 2
  }
}