Erdős–Borwein constant

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The Erdős–Borwein constant is the sum of the reciprocals of the Mersenne numbers. It is named after Paul Erdős and Peter Borwein.

By definition it is:

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

It can be proven 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 σ₀(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 resummed as such.

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