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.
