Erdös-Borwein constant

The Erdös-Borwein constant is the sum of the reciprocals of the Mersenne numbers.

By definition it is:

<math> E=\sum_{n=1}^{\infty}\frac{1}{2^n-1} \approx 1.60669 51524 15291 763... </math>

It can be proved that the following forms are equivalent to the former:

<math> E=\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}\frac{2^n+1}{2^n-1} </math>

<math> E=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{2^{mn}} </math>

<math> E=\sum_{n=1}^{\infty}\frac{\sigma_0(n)}{2^n} </math>

where <math>\sigma_0(n)</math> represents a multiplicative function, the number of positive divisors of the number <math>n</math>.

Paul Erdös showed that the constant E is an irrational number.