Erdős–Borwein constant

From Freepedia

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:

<math>

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

It can be proven 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=1+\sum_{n=1}^{\infty} \frac{1}{2^n(2^n-1)} </math>

<math>

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

where <math>\sigma_0(n)=d(n)</math> is the divisor function, a multiplicative function that equals the number of positive divisors of the number <math>n</math>. To prove the equivalence of these sums, note that they all take the form of Lambert series and can thus be resumed as such.

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