Family 024Number theoryNo LeanNot yet digested

An asymptotic formula for the number of totients

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)\to c

for every fixed c > 0, answering Erdős and Hall’s scaling question.

Manuscripts in this family · 1

Sep 25, 2026·An-asymptotic-formula-for-the-number-of-totients-September-25-2026.pdf

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

x\to\infty

for every fixed c > 0, answering a question of Erdős and Hall.

Full abstract