Constant of Erdős-Borwein

The constant of Erdős-Borwein is the nap Inverse S of the numbers of Mersenne.

By definition, it is equal to:

$$

E= \ sum_ {n=1} ^ {\ infty} \ frac {1} {2^n-1} \ approx 1,606 695.152.415 291.763…

It can be shown the equivalence of the following forms with the preceding formula:

$$

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 $\backslash \; sigma\_0\; (N)\; =d\; (N)\; \backslash ,$ is the Fonction divider, a multiplicative Fonction which is equal to the number of Diviseur S positive of the number $n\; \backslash ,$. To show the equivalence of these sums, let us note that they take all the form of a Série of Lambert and can thus be begun again like such.

Paul Erdős in 1948 showed that the constant E is a irrational Nombre.

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