{"id":"hexagon/2611.00116","pdf":"https://hexagonmath.org/pdf/2611.00116v1","url":"https://hexagonmath.org/abs/2611.00116v1","tags":["cs.CC","cs.DS","cs.LO","msc:11N05","msc:11Y16","msc:11A25","msc:68Q25","msc:68V20","Exponential Time Hypothesis","Mobius function"],"score":0,"title":"The parity of the prime-counting function in time $x^{o(1)}$","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Vaibhav Tulsyan"],"reviews":0,"subject":"math.NT","summary":"Computing the prime-counting function $\\pi(x)$ exactly takes $x^{1/2+o(1)}$ steps by the analytic method of Lagarias and Odlyzko, and to our knowledge its parity is known faster only on short intervals: Tao, Croot and Helfgott decide whether an interval of length at most $x^{1/2+c}$ near $x$ contains an odd number of primes in time $x^{1/2-c+o(1)}$. However, their identity needs the Mobius function $\\mu(d)$ for all $d\\le\\sqrt x$, and we know of no bound of the form $x^{o(1)}$ for the parity, even conditionally. We present deterministic algorithms that output $\\pi(x) \\bmod 2$ and the parity…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus"],"discussed":0,"published":"2026-11-05","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}