Editing Euler's totient function (section)

==Ratio of consecutive values==

In 1950 Somayajulu proved<ref name=Rib38>Ribenboim, p.38</ref><ref name=SMC16>Sándor, Mitrinović & Crstici (2006) p.16</ref>

:<math>\begin{align}
\lim\inf \frac{\varphi(n+1)}{\varphi(n)}&= 0 \quad\text{and} \\[5px]
\lim\sup \frac{\varphi(n+1)}{\varphi(n)}&= \infty.
\end{align}</math>

In 1954 [[Andrzej Schinzel|Schinzel]] and [[Wacław Sierpiński|Sierpiński]] strengthened this, proving<ref name=Rib38/><ref name=SMC16/> that the set

:<math>\left\{\frac{\varphi(n+1)}{\varphi(n)},\;\;n = 1,2,\ldots\right\}</math>

is [[Dense set|dense]] in the positive real numbers. They also proved<ref name=Rib38/> that the set

:<math>\left\{\frac{\varphi(n)}{n},\;\;n = 1,2,\ldots\right\}</math>

is dense in the interval (0,1).