Euler's Totient Function - Ratio of Consecutive Values

Ratio of Consecutive Values

In 1950 Somayajulu proved

$\lim\inf \frac{\varphi(n+1)}{\varphi(n)}= 0$ and $\lim\sup \frac{\varphi(n+1)}{\varphi(n)}= \infty.$

In 1954 Schinzel and Sierpiński strengthened this, proving that the set

$\left\{\frac{\varphi(n+1)}{\varphi(n)},\;\;n = 1,2,\cdots\right\}$

is dense in the positive real numbers. They also proved that the set

$\left\{\frac{\varphi(n)}{n},\;\;n = 1,2,\cdots\right\}$

is dense in the interval (0, 1).

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