Erdős–Tenenbaum–Ford constant

Short description: Mathematical 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 $N\to +\mathrm{\infty }$.

Tenenbaum improved the Erdős multiplication table estimate, and Ford established an asymptotic best bound.

External links

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