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
δ := 1 − 1 + log log 2 log 2 = 0.0860713320 … {\displaystyle \delta :=1-{\frac {1+\log \log 2}{\log 2}}=0.0860713320\dots }
where log {\displaystyle \log } is the natural logarithm. Following up on earlier work by Tenenbaum, Ford used this constant in analyzing the number H ( x , y , z ) {\displaystyle H(x,y,z)} of integers that are at most x {\displaystyle x} and that have a divisor in the range [ y , z ] {\displaystyle [y,z]} .
Multiplication table problem For each positive integer N {\displaystyle N} , let M ( N ) {\displaystyle M(N)} be the number of distinct integers in an N × N {\displaystyle N\times N} multiplication table. In 1960, Erdős studied the asymptotic behavior of M ( N ) {\displaystyle M(N)} and proved that
M ( N ) = N 2 ( log N ) δ + o ( 1 ) , {\displaystyle M(N)={\frac {N^{2}}{(\log N)^{\delta +o(1)}}},}
as N → + ∞ {\displaystyle N\to +\infty } . Tenenbaum improved the Erdős multiplication table estimate, and Ford established an asymptotic best bound.
References
External links Decimal digits of the Erdős–Tenenbaum–Ford constant on the OEIS