{"id":"openai-math/An-asymptotic-formula-for-the-number-of-totients","pdf":"https://github.com/openai/math/blob/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026/An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026","tags":["number-theory","euler-totient"],"score":0,"title":"An asymptotic formula for the number of totients","domain":"Number theory","family":"024","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"Let $`V(x)`$ count the distinct values of Euler's totient function up to x . We give an explicit asymptotic equivalent for $`V(x)`$. Its coefficient is a uniform limit of functions defined from finite arithmetic data. We also prove that $`V(cx)/V(x)\\to c`$ as $`x\\to\\infty`$ for every fixed c > 0, answering a question of Erdős and Hall.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"An asymptotic formula for the number of totients"}