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

{\displaystyle \delta

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 $N\times N$ multiplication table. In 1960,[5] 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

Luca, Florian; Pomerance, Carl (2014). "On the range of Carmichael's universal-exponent function" (PDF). Acta Arithmetica. 162 (3): 289–308. doi:10.4064/aa162-3-6. MR 3173026.
Tenenbaum, G. (1984). "Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné". Compositio Mathematica (in French). 51 (2): 243–263. MR 0739737.
Ford, Kevin (2008). "The distribution of integers with a divisor in a given interval". Annals of Mathematics. Second Series. 168 (2): 367–433. doi:10.4007/annals.2008.168.367. MR 2434882.
Koukoulopoulos, Dimitris (2010). "Divisors of shifted primes". International Mathematics Research Notices. 2010 (24): 4585–4627. arXiv:0905.0163. doi:10.1093/imrn/rnq045. MR 2739805. S2CID 7503281.
Erdős, Paul (1960). "An asymptotic inequality in the theory of numbers". Vestnik Leningrad. Univ. 15: 41–49. MR 0126424.

External links

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

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[1] Luca, Florian; Pomerance, Carl (2014). "On the range of Carmichael's universal-exponent function" (PDF). Acta Arithmetica. 162 (3): 289–308. doi:10.4064/aa162-3-6. MR 3173026.

[2] Tenenbaum, G. (1984). "Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné". Compositio Mathematica (in French). 51 (2): 243–263. MR 0739737.

[3] Ford, Kevin (2008). "The distribution of integers with a divisor in a given interval". Annals of Mathematics. Second Series. 168 (2): 367–433. doi:10.4007/annals.2008.168.367. MR 2434882.

[4] Koukoulopoulos, Dimitris (2010). "Divisors of shifted primes". International Mathematics Research Notices. 2010 (24): 4585–4627. arXiv:0905.0163. doi:10.1093/imrn/rnq045. MR 2739805. S2CID 7503281.

[5] Erdős, Paul (1960). "An asymptotic inequality in the theory of numbers". Vestnik Leningrad. Univ. 15: 41–49. MR 0126424.