Leikhman's Theorem (on the decomposition of Euler's totient values)

AbstractThis paper examines Euler’s totient function φ(n). It is establishedthat for any number ε that can be represented as ε = φ(n), there existdivisors σ, ω of n such that φ(σ) · φ(ω) = ε. Unlike the classicalmultiplicative property, which requires σ · ω = n and gcd(σ, ω) = 1,in this theorem σ and ω are arbitrary divisors of n, not necessarilycoprime and not necessarily multiplying to n; moreover, the equalityholds for all ε in a wide range of values. An empirical verificationis provided for all ε ≤ 10 000 000 using a computational experiment.The trivial cases σ = 1, ω = n and σ = n, ω = 1 are also allowed. Theresult represents a new observation on the structure of Euler’s totientvalues and may serve as a basis for further theoretical research.