The Erdős–Tenenbaum–Ford constant is a

mathematical constant A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Cons ...

that appears in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

. Named after mathematicians Paul Erdős,

Gérald Tenenbaum Gérald Tenenbaum is a French mathematician and novelist, born in Nancy on 1 April 1952.Kevin Ford, it is defined as :

\delta := 1 - \frac = 0.0860713320\dots

where

\log

is the

natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...

. 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

,z /math>.

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 mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essen ...

. In 1960, Erdős studied the

asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...

behavior of

M(N)

and proved that :

M(N) = \frac,

N \to +\infty

References

External links

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

OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to th ...

{{DEFAULTSORT:Erdős-Tenenbaum-Ford constant Mathematical constants Number theory