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

.

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Tags: Dirichlet Eta Function, L'Hôpital's Rule, Math, Monday Math, Riemann Zeta Function

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