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The Pólya–Vinogradov inequality with log log q for every character

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 1 / Oct 16, 2026 / CC BY 4.0

Abstract

Let χ\chi be a primitive Dirichlet character modulo q≥3q\ge3 and let M(χ)M(\chi) be the largest modulus of a partial sum ∑n≤xχ(n)\sum_{n\le x}\chi(n). The P\'olya--Vinogradov inequality bounds M(χ)M(\chi) by a constant times qlog⁡q\sqrt q\log q. Paley showed that qlog⁡log⁡q\sqrt q\log\log q is the most that can be hoped for, and Montgomery and Vaughan proved that this is the truth if the generalised Riemann hypothesis holds. Without any hypothesis we prove that M(χ)≤Cqlog⁡log⁡qM(\chi)\le C\sqrt q\log\log q for every primitive character χ\chi to every modulus q≥3q\ge3, with an absolute constant CC, and that ∣L′/L(1,χ)∣≤Clog⁡log⁡q|L'/L(1,\chi)|\le C\log\log q for every nonprincipal character. We also prove, from the elementary bounds of Chebyshev alone, that for every ε>0\varepsilon>0 there is a constant CεC_\varepsilon such that at most qεq^{\varepsilon} of the primitive characters modulo qq have M(χ)>Cεqlog⁡log⁡qM(\chi)>C_\varepsilon\sqrt q\log\log q, and at most qεq^{\varepsilon} of the nonprincipal characters have ∣L′/L(1,χ)∣>Cεlog⁡log⁡q|L'/L(1,\chi)|>C_\varepsilon\log\log q. The proof has two parts. The first is a transfer theorem for a single character: if θ<1\theta<1 is fixed and the LL-functions of the twists χψ\chi\psi of χ\chi by the characters ψ\psi of the moduli up to (log⁡q)16(\log q)^{16} have no zero in the rectangle ℜs>θ\Re s>\theta, ∣ℑs∣≤5q2|\Im s|\le5q^2, then M(χ)≤Cθqlog⁡log⁡qM(\chi)\le C_\theta\sqrt q\log\log q. It is proved by a contour shift for Riesz means of prime sums, a dissection of the circle into major and minor arcs, and a weak form of the Montgomery--Vaughan bound for exponential sums with multiplicative coefficients, which we prove with Selberg's sieve as the only arithmetic input. The second part counts the characters that fail the hypothesis of the transfer theorem by a zero-density theorem in the manner of Ingham and Montgomery, which bounds the number of zeros of all LL-functions to a modulus in such a rectangle by a power of the modulus that tends to zero as θ\theta tends to one. The same transfer theorem shows that a zero-free half-plane ℜs>θ\Re s>\theta for all Dirichlet LL-functions gives the bound for every character, and such a half-plane exists: OpenAI has proved that every Dirichlet LL-function, and every finite-order Hecke LL-function over Q(−3)\mathbb Q(\sqrt{-3}), has no zero with ℜs>11/12\Re s>11/12, by a mean square of twisted M\"obius sums over the Eisenstein integers, two Poisson summations, the transformation of Kubota's cubic theta function and the quadratic large sieve. Their proof is reproduced in full in a second appendix, with credit to them, so that the paper is complete in itself: the zero-density theorem is proved in the first appendix with explicit constants, and every statement of the paper is proved in the text.

Provenance statement

Deep Bhattacharjee chose the problem, set the requirements and reviewed the results. The proofs, the Lean 4 formalisation, the supporting computations and the text of the manuscript were developed with Claude (Anthropic), used through Claude Code. Theorems 1.1 and 1.3 and the exponential sum bound behind them are formalised in Lean 4 with Mathlib and checked by the Lean kernel. The formal proofs use only the standard axioms (propositional extensionality, choice and quotient soundness) and have no unproved steps. The one input not proved in the paper is the zero-free half-plane Re s > 7/8 for Dirichlet L-functions of nonprincipal characters. It comes from OpenAI's preprint "The quasi-Riemann hypothesis: a zero-free half-plane Re s > 7/8" (30 September 2026, https://github.com/openai/math), which has not been refereed. Its Lean proof was compiled from source and checked by the kernel together with ours, so the whole argument is machine-checked. The numerical steps of the proofs were also checked by independent programs in Python, Julia, C and C++, run by a shell script. The Lean files, the programs and the logs are archived at https://doi.org/10.5281/zenodo.23383716.
AI provenance filesAI_PROVENANCE_PolyaVinogradovLogLog.md · 3 KB

Tools used

OpenAI
ChatGPTVersion GPT SOL 6.1 Ultra

References

  1. Y. Ihara, V. K. Murty and M. Shimura, On the logarithmic derivatives of Dirichlet LL-functions at s=1s=1 , Acta Arith. 137 (2009), 253--276, doi:10.4064/aa137-3-6 (https://doi.org/10.4064/aa137-3-6).DOI
  2. H. L. Montgomery and R. C. Vaughan, Exponential sums with multiplicative coefficients , Invent. Math. 43 (1977), 69--82, doi:10.1007/BF01390204 (https://doi.org/10.1007/BF01390204).DOI
  3. OpenAI, The quasi-Riemann hypothesis: a zero-free half-plane ℜs>7/8\Re s>7/8 , preprint, 30 September 2026, with a Lean 4 formalisation, https://github.com/openai/math.link

Version history

  1. v1Submitted by Deep BhattacharjeeInitial depositSupersededOct 15, 2026
  2. v2Submitted by Deep BhattacharjeeSubmitted as a new versionSupersededOct 21, 2026
  3. v3Submitted by Deep BhattacharjeeSubmitted as a new versionCurrentOct 26, 2026