Erdős–Tenenbaum–Ford constant

The Erdős–Tenenbaum–Ford constant is a mathematical constant that appears in number theory.^[1] Named after mathematicians Paul Erdős, Gérald Tenenbaum, and Kevin Ford, it is defined as

${\displaystyle \delta$ :=1-{\frac {1+\log \log 2}{\log 2}}=0.0860713320\dots }

where ${\displaystyle \log }$ is the natural logarithm.

Following up on earlier work by Tenenbaum, Ford used this constant in analyzing the number ${\displaystyle H(x,y,z)}$ of integers that are at most ${\displaystyle x}$ and that have a divisor in the range ${\displaystyle [y,z]}$.^[2][3][4]

Multiplication table problem

For each positive integer ${\displaystyle N}$, let ${\displaystyle M(N)}$ be the number of distinct integers in an ${\displaystyle N\times N}$ multiplication table. In 1960,^[5] Erdős studied the asymptotic behavior of ${\displaystyle M(N)}$ and proved that

${\displaystyle M(N)={\frac {N^{2}}{(\log N)^{\delta +o(1)}}},}$

as ${\displaystyle N\to +\infty }$.