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).

Read more about this topic: Euler's Totient Function

Famous quotes containing the words ratio of, ratio and/or values:

“Personal rights, universally the same, demand a government framed on the ratio of the census: property demands a government framed on the ratio of owners and of owning.”
—Ralph Waldo Emerson (1803–1882)

“People are lucky and unlucky not according to what they get absolutely, but according to the ratio between what they get and what they have been led to expect.”
—Samuel Butler (1835–1902)

“With the breakdown of the traditional institutions which convey values, more of the burdens and responsibility for transmitting values fall upon parental shoulders, and it is getting harder all the time both to embody the virtues we hope to teach our children and to find for ourselves the ideals and values that will give our own lives purpose and direction.”
—Neil Kurshan (20th century)