{"id":"hexagon/2610.00449","pdf":"https://hexagonmath.org/pdf/2610.00449v1","url":"https://hexagonmath.org/abs/2610.00449v1","tags":["msc:11N05"],"score":0,"title":"A quantitative refinement of Lu's elementary prime number theorem","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Rajat Bhattacharjee"],"reviews":0,"subject":"math.NT","summary":"We refine Lu's elementary method for the prime number theorem to obtain, for every fixed $K>400$, \\[ \\psi(X)-X=O_K\\!\\left( X\\exp\\!\\left\\{-\\frac{\\sqrt{\\log X}}{(\\log\\log X)^K}\\right\\}\\right) \\qquad(X\\to\\infty), \\] where $\\psi$ is Chebyshev's function. The main ingredient is a comparison estimate that remains uniform as the degree grows and the short-interval parameter shrinks. Combining this estimate with exponential weights in Balog's identity, a single finite energy step and positivity-based interpolation yields the stated bound, retaining the full degree tail and all arithmetic correction…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-22","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}