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
- [math]\displaystyle{ \delta := 1 - \frac{1 + \log \log 2}{\log 2} = 0.0860713320\dots }[/math]
where [math]\displaystyle{ \log }[/math] is the natural logarithm.
Following up on earlier work by Tenenbaum, Ford used this constant in analyzing the number [math]\displaystyle{ H(x,y,z) }[/math] of integers that are at most [math]\displaystyle{ x }[/math] and that have a divisor in the range [math]\displaystyle{ [y,z] }[/math].[2][3][4]
Multiplication table problem
For each positive integer [math]\displaystyle{ N }[/math], let [math]\displaystyle{ M(N) }[/math] be the number of distinct integers in an [math]\displaystyle{ N \times N }[/math] multiplication table. In 1960,[5] Erdős studied the asymptotic behavior of [math]\displaystyle{ M(N) }[/math] and proved that
- [math]\displaystyle{ M(N) = \frac{N^2}{(\log N)^{\delta + o(1)}}, }[/math]
as [math]\displaystyle{ N \to +\infty }[/math].
References
- ↑ Luca, Florian; Pomerance, Carl (2014). "On the range of Carmichael's universal-exponent function". Acta Arithmetica 162 (3): 289–308. doi:10.4064/aa162-3-6. https://math.dartmouth.edu/~carlp/rangeoflambda13.pdf.
- ↑ Tenenbaum, G. (1984). "Sur la probabilité qu'un entier possède un diviseur dans un intervalle donné" (in French). Compositio Mathematica 51 (2): 243–263. https://www.numdam.org/item?id=CM_1984__51_2_243_0.
- ↑ 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.
- ↑ Koukoulopoulos, Dimitris (2010). "Divisors of shifted primes". International Mathematics Research Notices 2010 (24): 4585–4627. doi:10.1093/imrn/rnq045.
- ↑ Erdős, Paul (1960). "An asymptotic inequality in the theory of numbers". Vestnik Leningrad. Univ. 15: 41–49.
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