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

where is the natural logarithm.

Following up on earlier work by Tenenbaum, Ford used this constant in analyzing the number of integers that are at most and that have a divisor in the range .[2][3][4]

Multiplication table problem

For each positive integer , let be the number of distinct integers in an multiplication table. In 1960,[5] Erdős studied the asymptotic behavior of and proved that

as .

References

  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.
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