Erdős Problem 416: does $V(2x)/V(x)\to2$ for the number $V(x)$ of totient values up to $x$, and is there an asymptotic formula?

The paper gives an explicit asymptotic equivalent $V(x)\sim\frac{x}{\log x}G(x)A(\theta(x))$, where the bounded positive coefficient $A$ depends on a phase $\theta(x)\in[0,1)$ and is a uniform limit of functions defined from finite prime and integer data, and proves $V(cx)/V(x)\to c$ for every fixed $c>0$, answering Erdős-Hall. It also gives asymptotics for totients whose least preimage lies in $(kx,(k+1)x]$, a further question of Erdős (1979), with a positive coefficient for $k=1,2$ and identically zero counts when no totient $d$ has least preimage above $kd$. Not shown: a closed-form constant (the coefficient is a limit, not a number), or secondary terms.

The paper gives an explicit asymptotic equivalent $V(x)\sim\frac{x}{\log x}G(x)A(\theta(x))$, where the bounded positive coefficient $A$ depends on a phase $\theta(x)\in[0,1)$ and is a uniform limit of functions defined from finite prime and integer data, and proves $V(cx)/V(x)\to c$ for every fixed $c>0$, answering Erdős-Hall. It also gives asymptotics for totients whose least preimage lies in $(kx,(k+1)x]$, a further question of Erdős (1979), with a positive coefficient for $k=1,2$ and identically zero counts when no totient $d$ has least preimage above $kd$. Not shown: a closed-form constant (the coefficient is a limit, not a number), or secondary terms.