Erdős-Borwein constant

The Erdős – Borwein constant , named after Paul Erdős and Peter Borwein , is a mathematical constant . It is defined as the sum of the reciprocal values ​​of the Mersenne numbers :

${\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}=1{,}60669{\text{ }}51524{\text{ }}15291{\text{ }}76378{\text{ }}...}$(Follow A065442 in OEIS )

The following representations are equivalent to this:

${\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}{\frac {2^{n}+1}{2^{n}-1}}}$

${\displaystyle E=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}}$

${\displaystyle E=1+\sum _{n=1}^{\infty }{\frac {1}{2^{n}(2^{n}-1)}}}$

${\displaystyle E=\sum _{n=1}^{\infty }{\frac {\sigma _{0}(n)}{2^{n}}}}$

where σ ₀ ( n ) = d ( n ) is the number of divisors (number of positive divisors of n ). To prove the equivalence, note that all sums can be expressed as Lambert series and then summed up.

The constant was already considered by Euler in 1749 . Erdős showed in 1948 that E is an irrational number . Borwein showed in 1992 that in general

${\displaystyle \sum _{n=1}^{\infty }{\frac {1}{q^{n}-r}}}$     and     ${\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{q^{n}-r}}}$

for every integer q ≠ 0, ± 1 and every rational number r ≠ 0, q ^{n are} irrational but not liouvillian .

literature

• Steven R. Finch: Mathematical constants . Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , pp. 355 and 357 (English)

Web links

• Eric W. Weisstein : Erdős-Borwein Constant . In: MathWorld (English).
• Follow A038631 in OEIS ( continued fraction expansion of e )

Individual evidence

1. ↑ Leonhard Euler : Consideratio quarumdam serierum, quae singularibus proprietatibus sunt praeditae . (June 19, 1749/26 January 1750), Novi commentarii academiae scientiarum Petropolitanae 3, 1753, pp. 86-108 (Latin; " s = 1.606695152415291" on p. 108 )
2. ^ Paul Erdős : On arithmetical properties of Lambert series (July 8, 1948). In: The Journal of the Indian Mathematical Society , 12, 1948, pp. 63–66 (English)
3. ↑ Peter B. Borwein : On the irrationality of certain series . (PDF; 3.3 MB) December 11, 1991. In: Mathematical Proceedings of the Cambridge Philosophical Society , 112, 1992, pp. 141–146 (English)