OpenAI research collection · 1 manuscript · Source checked 2026-10-19
Gives an asymptotic equivalent for the number V(x) of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, V(cx)/V(x)→ c for every fixed c > 0, answering Erdős and Hall’s scaling question.
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
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)→ c as x→∞ for every fixed c > 0, answering a question of Erdős and Hall.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.
Explore the AGI Clock ↗