Erdős–Tenenbaum–Ford constant explained

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

\delta : = 1 - (1+loglog2)/(log2) = 0 . 0860713320... 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] .^{[2] [3] [4]}

Multiplication table problem

For each positive integer

N , let M(N) be the number of distinct integers in an N x N multiplication table. In 1960,^[5] Erdős studied the asymptotic behavior of M(N) and proved thatM(N) = (N ^{2})/(( log N) ^{\delta}) , as N \to +inf ty .

External links

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

Notes and References

1. Florian. Luca. Carl. Pomerance. Carl Pomerance. On the range of Carmichael's universal-exponent function. Acta Arithmetica. 162. 2014. 3. 289–308. 3173026. 10.4064/aa162-3-6.
2. Tenenbaum. G.. Gérald Tenenbaum. 2. Compositio Mathematica. French. 739737. 243–263. Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné. 51. 1984.
3. Ford. Kevin. Kevin Ford (mathematician). 10.4007/annals.2008.168.367. 2. Annals of Mathematics. 2434882. 367–433. Second Series. The distribution of integers with a divisor in a given interval. 168. 2008. free. math/0401223.
4. Koukoulopoulos. Dimitris. 0905.0163. 10.1093/imrn/rnq045. 24. International Mathematics Research Notices. 2739805. 4585–4627. Divisors of shifted primes. 2010. 2010. 7503281.
5. Paul. Erdős. Paul Erdős. An asymptotic inequality in the theory of numbers. Vestnik Leningrad. Univ.. 15. 1960. 41–49. 0126424.