Pith Number

pith:MDKI5BBZ

pith:2024:MDKI5BBZKUVI2OCLB44XAKO3AI

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An additive application of the resonance method

Athanasios Sourmelidis

The resonance method applied additively to trigonometric polynomials with positive Fourier coefficients improves Omega results and yields sharper bounds for the divisor and circle problems.

arxiv:2411.14221 v2 · 2024-11-21 · math.NT

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Claims

C1strongest claim

We improve upon an Omega result due to Soundararajan with respect to general trigonometric polynomials having positive Fourier coefficients... this leads to better extreme results in lattice point problems such as Dirichlet's divisor problem and Gauss' circle problem.

C2weakest assumption

The resonance method can be transferred from its prior multiplicative settings to the additive setting of trigonometric polynomials with positive coefficients while preserving the claimed quantitative improvement over Dirichlet approximation.

C3one line summary

Resonance method applied to positive-coefficient trigonometric polynomials improves Omega bounds and strengthens results in Dirichlet's divisor and Gauss' circle problems.

References

20 extracted · 20 resolved · 0 Pith anchors

[1] Lower bounds for the maximum of the Riemann zeta function along vertical lines.Math 2015 · doi:10.1007/s00208-015-1290-0

[2] Large values ofL-functions from the Selberg class.J 2017 · doi:10.1016/j.jmaa.2016.08.044

[3] C. Aistleitner, K. Mahatab, and M. Munsch. Extreme values of the Riemann zeta function on the 1-line. Int. Math. Res. Notices, 2019(22):6924–6932, 2019. doi: 10.1093/imrn/rnx331 2019 · doi:10.1093/imrn/rnx331

[4] One more proof of Kronecker’s theorem.J 1932 · doi:10.1112/jlms/s1-7.4.274

[5] Large greatest common divisor sums and extreme values of the Riemann zeta function.Duke Math 2017 · doi:10.1215/00127094-0000005x

Cited by

1 paper in Pith

Sharp omega results for the divisor and circle problems

Receipt and verification

First computed	2026-05-27T01:05:32.855997Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

60d48e8439552a8d384b0f397029db0214353d377a2370ec21b5184025e7053a

Aliases

arxiv: 2411.14221 · arxiv_version: 2411.14221v2 · doi: 10.48550/arxiv.2411.14221 · pith_short_12: MDKI5BBZKUVI · pith_short_16: MDKI5BBZKUVI2OCL · pith_short_8: MDKI5BBZ

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/MDKI5BBZKUVI2OCLB44XAKO3AI \
  | 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: 60d48e8439552a8d384b0f397029db0214353d377a2370ec21b5184025e7053a

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "92c19bbd23228a63c3aef234725eea0945f80d12ef6ae8a9db788968812ac226",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2024-11-21T15:29:18Z",
    "title_canon_sha256": "2a96b7bbcb5fcdadc23232a7981818986d5b3fbf11e06237805de6d69935cf5e"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2411.14221",
    "kind": "arxiv",
    "version": 2
  }
}