On an Erdős–Straus type problem

Motivated by the Erdős–Straus conjecture, we study whether the Euler totient ratio <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mfrac> <m:mrow> <m:mi>φ</m:mi> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mi>n</m:mi> </m:mfrac> </m:math> φ(n)/n can be expressed as a sum of three unit fractions, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mfrac> <m:mrow> <m:mi>φ</m:mi> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>)</m:mo> </m:mrow> <m:mi>n</m:mi> </m:mfrac> <m:mo>=</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>x</m:mi> </m:mfrac> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>y</m:mi> </m:mfrac> <m:mo>+</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mi>z</m:mi> </m:mfrac> <m:mo>,</m:mo> </m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo>,</m:mo> <m:mi>z</m:mi> <m:mo>∈</m:mo> <m:mi>ℕ</m:mi> <m:mo>.</m:mo> </m:mrow> </m:math> φ(n)/n = 1/x + 1/y + 1/z, x, y, z ∈ N. We prove that such a representation is impossible for prime powers p k with p ≥ 11, and for integers of the form p k 1 q k 2 with p ≥ 5 and q ≥ 15 p . In contrast, for 2 ≤ p ≤ 7 the representation exists for all p k . Moreover, for every integer r ≥ 3, there are infinitely many integers n with ω ( n )= r for which no such representation exists. Further, if a representation holds, we obtain the bounds <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block"> <m:mrow> <m:mn>2</m:mn> <m:mo>≤</m:mo> <m:mi>x</m:mi> <m:mo>≤</m:mo> <m:mn>9</m:mn> <m:mo>+</m:mo> <m:mn>3</m:mn> <m:msup> <m:mi>e</m:mi> <m:mi>γ</m:mi> </m:msup> <m:mtext>log log</m:mtext> <m:mi>n</m:mi> <m:mo>,</m:mo> <m:mspace width="1em"/> <m:mfrac> <m:mrow> <m:mi>n</m:mi> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>φ</m:mi> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>)</m:mo> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mfrac> <m:mo>≤</m:mo> <m:mi>y</m:mi> <m:mo>≤</m:mo> <m:mfrac> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>φ</m:mi> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>)</m:mo> <m:mi>x</m:mi> <m:mo>−</m:mo> <m:mi>n</m:mi> </m:mrow> </m:mfrac> <m:mo>.</m:mo> </m:mrow> </m:math> 2 ≤ x ≤ 9 + 3eγ log log n, nx/φ(n)x - n ≤ y ≤ 2nx/φ(n)x - n. In addition, if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>≥</m:mo> <m:mfrac> <m:mrow> <m:mn>3</m:mn> <m:mo>(</m:mo> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> <m:mo>)</m:mo> <m:mo>(</m:mo> <m:msup> <m:mi>e</m:mi> <m:mi>γ</m:mi> </m:msup> <m:mtext>log log</m:mtext> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>3</m:mn> <m:mo>)</m:mo> </m:mrow> <m:mi>n</m:mi> </m:mfrac> </m:mrow> </m:math> x ≥ {3(n-1)(eγ log log n + 3)}{n} , then rad <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow>