{"id":"hexagon/2610.00256","pdf":"https://hexagonmath.org/pdf/2610.00256v2","url":"https://hexagonmath.org/abs/2610.00256v2","tags":["math.CA","msc:11L40","msc:11M06","msc:11M20","msc:11L07","msc:11N13","Euler--Kronecker constants","P\\'olya--Vinogradov inequality","Selberg sieve","character sums"],"score":0,"title":"The P\\'olya--Vinogradov inequality with $\\log\\log q$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Deep Bhattacharjee","Ushashi Bhattacharya"],"reviews":0,"subject":"math.NT","summary":"We prove that every primitive Dirichlet character $\\chi$ modulo $q\\ge3$ has character sums $\\sum_{n\\le x}\\chi(n)$ of size at most $C\\sqrt q\\log\\log q$, with an absolute constant $C$. 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 $\\Re s>7/8$ for Dirichlet $L$-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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code"],"discussed":0,"published":"2026-10-16","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"some"}