Condition W3: Absence of Siegel Zeros for Characters Modulo 3^K via the CM Structure of Z[ω]
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2026
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| _version_ | 1866901062375440384 |
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| author | Wakil, Khayyam |
| author_facet | Wakil, Khayyam |
| contents | <p>We prove Condition W3: for every primitive Dirichlet character χ modulo <em>q</em> = 3<sup><em>K</em></sup>, the <em>L</em>-function <em>L</em>(<em>s</em>, χ) has no real zero in the region σ > 1 − <em>c</em>/log <em>q</em> for an absolute, effectively computable constant <em>c</em> > 0. This closes the gap marked [NOTE] in the companion paper [4] and renders the main theorem of that paper — level of distribution θ<sub>W</sub> = 5/8 for cascade moduli — unconditional.</p> <p>The proof has three steps, each using the same algebraic structure. <strong>Step 1 (Lifting).</strong> Every primitive character χ (mod 3<sup><em>K</em></sup>) lifts canonically to a primitive Hecke Grössencharacter Ψ<sub>χ</sub> of K = Q(ω) = Q(√−3) of conductor <sup>2<em>K</em></sup>, where = (1−ω) is the unique prime of Z[ω] above 3. <strong>Step 2 (Stark–Baker).</strong> Since K is a CM field of class number 1, the Stark–Baker theorem excludes Siegel zeros for <em>L</em>(<em>s</em>, Ψ<sub>χ</sub>) in an effective zero-free region. <strong>Step 3 (Transfer).</strong> The factorisation <em>L</em>(<em>s</em>, Ψ<sub>χ</sub>) = <em>L</em>(<em>s</em>, χ)·<em>L</em>(<em>s</em>, χε) transfers the zero-free region to <em>L</em>(<em>s</em>, χ).</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19016777 |
| institution | Zenodo |
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| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Condition W3: Absence of Siegel Zeros for Characters Modulo 3^K via the CM Structure of Z[ω] Wakil, Khayyam Siegel Zeros Dirichlet L-Functions CM Fields Hecke Grössencharacter Stark-Baker Theorem Cascade Moduli Ramification Theory Eisenstein Integers Zero-Free Regions Level of Distribution <p>We prove Condition W3: for every primitive Dirichlet character χ modulo <em>q</em> = 3<sup><em>K</em></sup>, the <em>L</em>-function <em>L</em>(<em>s</em>, χ) has no real zero in the region σ > 1 − <em>c</em>/log <em>q</em> for an absolute, effectively computable constant <em>c</em> > 0. This closes the gap marked [NOTE] in the companion paper [4] and renders the main theorem of that paper — level of distribution θ<sub>W</sub> = 5/8 for cascade moduli — unconditional.</p> <p>The proof has three steps, each using the same algebraic structure. <strong>Step 1 (Lifting).</strong> Every primitive character χ (mod 3<sup><em>K</em></sup>) lifts canonically to a primitive Hecke Grössencharacter Ψ<sub>χ</sub> of K = Q(ω) = Q(√−3) of conductor <sup>2<em>K</em></sup>, where = (1−ω) is the unique prime of Z[ω] above 3. <strong>Step 2 (Stark–Baker).</strong> Since K is a CM field of class number 1, the Stark–Baker theorem excludes Siegel zeros for <em>L</em>(<em>s</em>, Ψ<sub>χ</sub>) in an effective zero-free region. <strong>Step 3 (Transfer).</strong> The factorisation <em>L</em>(<em>s</em>, Ψ<sub>χ</sub>) = <em>L</em>(<em>s</em>, χ)·<em>L</em>(<em>s</em>, χε) transfers the zero-free region to <em>L</em>(<em>s</em>, χ).</p> |
| title | Condition W3: Absence of Siegel Zeros for Characters Modulo 3^K via the CM Structure of Z[ω] |
| topic | Siegel Zeros Dirichlet L-Functions CM Fields Hecke Grössencharacter Stark-Baker Theorem Cascade Moduli Ramification Theory Eisenstein Integers Zero-Free Regions Level of Distribution |
| url | https://doi.org/10.5281/zenodo.19016777 |