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Erdös-Borwein constant

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

By definition it is:

$E=\sum_{n=1}^{\infty}\frac{1}{2^n-1} \approx 1.60669 51524 15291 763...$

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

$E=\sum_{n=1}^{\infty}\frac{1}{2^{n^2}}\frac{2^n+1}{2^n-1}$

$E=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{2^{mn}}$

$E=\sum_{n=1}^{\infty}\frac{\sigma_0(n)}{2^n}$

where $\sigma_0(n)$ represents a multiplicative function, the number of positive divisors of the number $n$.

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

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