Publications & Preprints

On the Quantitative Sieve Parity Bypass under the Siegel Zero Hypothesis

This paper develops a quantitative form of the “exceptional-character bypass” of the sieve parity barrier for the binary Goldbach problem. The main statement is explicitly conditional: if a Siegel zero ${\beta }_1=1-{\delta }_1$ of $L(s,{\chi }_D)$ exists, then the number of representations satisfies $r(N)>0$ for all even $N$ above an explicit threshold depending on ${\delta }_1$.

Status correction (May 2026). A later source audit found that the draft’s derivation additionally rests on a broad Hardy–Littlewood-type dominance assumption that has not been matched to any published theorem. The claimed bypass of the parity barrier is therefore not established from the Siegel-zero hypothesis alone; the text is kept online as a documented draft, with its status corrected here and on the results page. The paper does not prove the Goldbach conjecture.

[PDF]

@misc{zheleznov2026siegel,
  author       = {Zheleznov, Sergey},
  title        = {On the Quantitative Sieve Parity Bypass
                  under the Siegel Zero Hypothesis},
  year         = {2026},
  month        = may,
  howpublished = {Unreviewed draft v1, goldbach.zheleznov.com},
  url          = {https://goldbach.zheleznov.com/papers/paper1_bk_plus.pdf},
  note         = {Accessed YYYY-MM-DD}
}

The Bilinear Minor-Arcs Hypothesis as the Unifying Isomorphism of the Sieve Parity Barrier

A meta-theoretic survey. Ten historically distinct formulations of the parity barrier — from classical sieve-theoretic ones to statements in the language of bilinear sums — are presented as projections of a single hypothesis, which the author calls the Bilinear Minor-Arcs Hypothesis (BMAH).

The correspondence between these formulations is offered as a working dictionary: no equivalence theorem is claimed, the supporting arguments are sketches, and the BMAH itself remains a conjecture. The text should be read as a map of the landscape, not as a summary of established facts.

[PDF]

@misc{zheleznov2026bmah,
  author       = {Zheleznov, Sergey},
  title        = {The Bilinear Minor-Arcs Hypothesis as the
                  Unifying Isomorphism of the Sieve Parity Barrier},
  year         = {2026},
  month        = may,
  howpublished = {Unreviewed draft v1, goldbach.zheleznov.com},
  url          = {https://goldbach.zheleznov.com/papers/paper2_parity_dictionary.pdf},
  note         = {Accessed YYYY-MM-DD}
}

A Balanced Divisor-Weight Orthogonality Problem Arising from Additive Goldbach Witnesses

This note derives exact second-moment identities for shifted Möbius sums — sums of the shape ${\sum }_{n\le x}\mu (n)\mu (n+h)$ — arising from additive Goldbach witnesses. Eight precise statements are proved. The central estimate ${\mathrm{C}\mathrm{W}\mathrm{C}}_3$ is stated in the note as a conjecture; since May 2026 it has additionally been certified in the project’s Lean layer as a theorem modulo explicit classical inputs. It is not used as an input for any claim about the binary Goldbach problem.

[PDF]

@misc{zheleznov2026cwc3,
  author       = {Zheleznov, Sergey},
  title        = {A Balanced Divisor-Weight Orthogonality Problem
                  Arising from Additive Goldbach Witnesses},
  year         = {2026},
  month        = apr,
  howpublished = {Archival draft (status box revised 2026-05-25),
                  goldbach.zheleznov.com},
  url          = {https://goldbach.zheleznov.com/papers/cwc3_note.pdf},
  note         = {Accessed YYYY-MM-DD}
}