The investigation of Euler's totient function preimages

Abstract

We propose a lower estimation for computing quantity of the inverses of Euler's function. We answer the question about the multiplicity of $m$ in the equation $\phi (x)=m$ \cite{Ford}. An analytic expression for exact multiplicity of $m=2^{2^n+a}$, where $a\in N$, $a<2^n$, $\phi (t)=2^{2^n+a}$ was obtained. A lower bound of inverses number for arbitrary $m$ was found. New numerical metric was proposed.

"Sixth International Conference on Analytic Number Theory and 11 Spatial Tessellations. Voronoy Conference"