Erdős–Tenenbaum–Ford constant

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

where $\log$ is the natural logarithm.

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,[2] Erdős studied the asymptotic behavior of $M(N)$ and proved that

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

as $N\to +\infty$ .

References

Florian Luca and Carl Pomerance (2014). "On the range of Carmichael's universal-exponent function". Acta Arithmetica. 162. pp. 289–308.CS1 maint: uses authors parameter (link)
Paul Erdős (1960). "An asymptotic inequality in the theory of numbers". Vestnik Leningrad. Univ. 15. pp. 41–49.

External links

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Florian Luca and Carl Pomerance (2014). "On the range of Carmichael's universal-exponent function". Acta Arithmetica. 162. pp. 289–308.CS1 maint: uses authors parameter (link)

[2] Paul Erdős (1960). "An asymptotic inequality in the theory of numbers". Vestnik Leningrad. Univ. 15. pp. 41–49.