## Monday Math 133

Find the limit  as a function of positive, real x; where ζ(s) is the Riemann zeta function.

The Riemann zeta function has a pole at s=1, while the term in the brackets is zero at s=1. Recall from here that the Dirichlet eta function η(s), which is defined for all complex numbers s, is related to the Riemann zeta function by
,
and thus
.
Therefore,
,
and since η(1)=ln2, we then see
.
This last limit has indeterminate form 0/0, and so is amenable to L’Hôpital’s rule:
,
and so
.