v4.0.0: The Pólya–Vinogradov inequality with log log q for every character

Deep Bhattacharjee and Ushashi Bhattacharya, The Pólya–Vinogradov inequality with log log q for every character. Version 4, single column (amsart), 53 pages.

M(χ) ≤ C √q log log q for every primitive Dirichlet character χ modulo q ≥ 3, with an absolute
constant C, where M(χ) is the largest modulus of a partial sum of χ, and |L'/L(1, χ)| ≤ C log log q
for every nonprincipal character (Theorem 1.5). The proof combines a transfer theorem for a single
character with the zero-free half-plane Re s > 11/12 for all Dirichlet L-functions; the half-plane
and its proof are due to OpenAI, and Appendix B gives the proof in full, with every external input
a published theorem.

New in this version:

• Section 8: the Euler–Kronecker constant γ_m of Q(ζ_m) satisfies γ_m ≤ 12 log m + 36 log log m + C
  for every m ≥ 3, an order known before only under the generalised Riemann hypothesis (Ihara), and
  γ_q ≥ −C log q log log q for primes q (Theorem 1.6).
• No statement rests on an unrefereed source: the preprints of OpenAI are cited for attribution
  only.
• A shorter abstract and the standard structure (no note under the abstract); the new affiliation
  of the first author.
• checks/python/eisenstein_check.py checks the Gauss-sum and reciprocity identities of Lemma B.3
  over the Eisenstein integers.

Version 3 (10.5281/zenodo.23672710) is hexagon:2610.00256v3; this version replaces it there.

Files: the paper (PolyaVinogradovLogLog_final2.pdf), its LaTeX source with the figures
(PolyaVinogradovLogLog_final2-tex.zip) and the arXiv tarball
(PolyaVinogradovLogLog_final2-arxiv.tar.gz).

Contact: d.bhattacharjee@erl-forschung.de, itsdeep@live.com; for this repository egspl@proton.me.