HEXAGONbeta

The P\'olya--Vinogradov inequality with log⁡log⁡q\log\log q

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.CA — Classical Analysis and ODEs

Contributed by Deep Bhattacharjee, Ushashi Bhattacharya ↗

Version 2 / Oct 16, 2026 / CC BY 4.0

Abstract

We prove that every primitive Dirichlet character χ\chi modulo q≥3q\ge3 has character sums ∑n≤xχ(n)\sum_{n\le x}\chi(n) of size at most Cqlog⁡log⁡qC\sqrt q\log\log q, with an absolute constant CC. Montgomery and Vaughan proved this bound under the generalised Riemann hypothesis, and Paley showed that it is optimal. The only input from outside the paper is the zero-free half-plane ℜs>7/8\Re s>7/8 for Dirichlet LL-functions. On the major arcs this half-plane gives a power saving for twisted prime sums; on the minor arcs we prove a weak form of the Montgomery--Vaughan bound for exponential sums with multiplicative coefficients, with Selberg's sieve as the only arithmetic input. The same half-plane gives ∣L′/L(1,χ)∣≪log⁡log⁡q|L'/L(1,\chi)|\ll\log\log q for every nonprincipal χ\chi and, for every prime qq, the two-sided bound −Clog⁡qlog⁡log⁡q≤γq≤Clog⁡q-C\log q\log\log q\le\gamma_q\le C\log q for the Euler--Kronecker constant of the cyclotomic field Q(ζq)\mathbb Q(\zeta_q).

Provenance statement

The proofs, the Lean 4 formalisation, the checking programs and the text were drafted with Claude (Anthropic) through Claude Code under the direction of Deep Bhattacharjee, who ran the computations and the Lean build. Every proof is formalised in Lean 4 (Lean 4.34.1, Mathlib v4.34.1) and compiled together with the unchanged Lean sources of OpenAI's zero-free half-plane Re s > 7/8; the main theorems depend only on propext, Classical.choice and Quot.sound, with no sorry. The half-plane is OpenAI's unrefereed preprint of 30 September 2026 and is the one result used without proof. Release v1.1.2: https://github.com/creelie/Breakthrough-problems-3/releases/tag/v1.1.2. Both authors approved the final version.
AI provenance filesAI-PROVENANCE-HEXAGON.md · 2 KB

Tools used

Anthropic
Claude CodeVersion Opus 5.5 Ultracode

References

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  2. D. Bhattacharjee and U. Bhattacharya, The Pólya--Vinogradov inequality with log⁡log⁡q\log\log q: Lean 4 formalisation and verification code , GitHub release v1.1.2, 2026, .
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  1. OpenAI, The quasi-Riemann hypothesis: a zero-free half-plane ℜs>7/8\Re s>7/8 , preprint (not refereed), 30 September 2026, with a Lean 4 formalisation, repository https://github.com/openai/mathlink
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Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 15, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 21, 2026