Let denote the – th harmonic number. Prove that

where denotes the Catalan’s constant.

**Solution**

We begin with a lemma:

* Lemma: *Let denote the dilogarithm function. It holds that

*Proof: *It is well known that

(1)

Setting we have that:

Setting we have that:

However,

where is the inverse tangent function. It now follows that

in view of the well known series .

Substracting the above relations we get the proof of the lemma.

* Theorem: *Let . It holds that

Setting we have that . The result now follows immediately using the lemma above as well as the fact that .