Erdős–Tenenbaum–Ford constant

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

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

where $\log$ is the natural logarithm.

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

Multiplication table problem

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

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

as $N\to +\infty$.

External links

• Decimal digits of the Erdős–Tenenbaum–Ford constant on the OEIS