Erdős–Borwein constant

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

By definition it is:

$E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}\approx 1.606695152415291763...$

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

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

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

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

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

where $\sigma _{0}(n)=d(n)$ is the divisor function, a multiplicative function that equals the number of positive divisors of the number $n$. To prove the equivalence of these sums, note that they all take the form of Lambert series and can thus be resumed as such.

Paul Erdős in 1948 showed that the constant E is an irrational number.