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:

The following representations are equivalent to:

Where σ0 (n) = d ( n) is the divisor number is (number of positive divisors of n ). To demonstrate the equivalence, Note that, all the sums are expressed as a Lambert- rows and umsummiert.

The constant has been considered already in 1749 by Euler. Erdős showed in 1948 that e is an irrational number. Borwein showed in 1992 that generally

For any integer q ≠ 0, ± 1 and each rational number r ≠ 0, qn irrational, but are not liouvillesch.