Pith Number

pith:GBN5RVHG

pith:2026:GBN5RVHGBICXGVVK44FBZF57MG

not attested not anchored not stored refs resolved

Large values of shifted mixed character sums

N\'eo Tardy

For non-principal characters modulo a prime, incomplete mixed sums have maximum size between √p log log p and √p log p.

arxiv:2605.13715 v1 · 2026-05-13 · math.NT · math.CA

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\usepackage{pith}
\pithnumber{GBN5RVHGBICXGVVK44FBZF57MG}

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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

✓ Portable graph bundle live · download bundle · merged state

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

√p log log p ≪ max_{0 ≤ θ < 1} |F_χ(α,β;θ)| ≪ √p log p for non-principal χ mod prime p.

C2weakest assumption

The character χ is non-principal and p is an odd prime; the proof assumes standard analytic continuation and zero-free regions or Polya-Vinogradov-type inequalities that are extended to the incomplete mixed setting.

C3one line summary

For non-principal χ mod prime p, the max over θ of |sum_{αp < n ≤ βp} χ(n) e(nθ)| satisfies √p log log p ≪ max ≪ √p log p.

References

15 extracted · 15 resolved · 0 Pith anchors

[1] Moments of polynomials with random multiplicative coefficients 2022

[2] Distribution of mixed character sums and extremal problems for Littlewood polynomials 2025

[3] Explicit merit factor formulae for Fekete and Turyn polynomials 2002

[4] Zeros of Fekete polynomials 2000

[5] Upper bounds for theLq norm of Fekete polynomials on subarcs 2012

Formal links

1 machine-checked theorem link

Receipt and verification

First computed	2026-05-18T02:44:16.712886Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

305bd8d4e60a057356aae70a1c97bf618664bee634f1f9e990adab9d05f39995

Aliases

arxiv: 2605.13715 · arxiv_version: 2605.13715v1 · doi: 10.48550/arxiv.2605.13715 · pith_short_12: GBN5RVHGBICX · pith_short_16: GBN5RVHGBICXGVVK · pith_short_8: GBN5RVHG

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/GBN5RVHGBICXGVVK44FBZF57MG \
  | 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: 305bd8d4e60a057356aae70a1c97bf618664bee634f1f9e990adab9d05f39995

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7c5e4c0f3ed51a3c0686beae410510a32c295a8b6ddfa0623aa02c6eefc5a5cc",
    "cross_cats_sorted": [
      "math.CA"
    ],
    "license": "http://creativecommons.org/licenses/by-sa/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-05-13T16:00:49Z",
    "title_canon_sha256": "a77cbed095917448ea430c685516d7a1d4f98649f63cbed9dbafc1d2f65ea4f2"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13715",
    "kind": "arxiv",
    "version": 1
  }
}