How does the Uniform Limit Theorem prove that the Riemann zeta function is analytic on Re(s) > 1?

Conditions

• $f_n(s)$ are the partial sums of the zeta series.
• The Uniform Limit Theorem applies to sequences of analytic functions converging uniformly on a domain.
• $a$ is an arbitrarily small positive constant ($a>0$).

Reasoning, step by step

1. Identify that each term $k^{-s}$ and thus each finite partial sum $f_n(s)$ is an analytic function.
2. Recall the established result that $f_n(s)$ converges uniformly to $\zeta(s)$ on the region $\operatorname{Re}(s)\ge 1+a$.
3. Apply the Uniform Limit Theorem to conclude that $\zeta(s)$ is analytic on $\operatorname{Re}(s)\ge 1+a$.
4. Argue that because $a>0$ can be arbitrarily small, the property of analyticity extends to the entire open half-plane $\operatorname{Re}(s)>1$.

Example

The slide states: 'Therefore $\zeta(s)$ is analytic on Re(s) $\ge 1 + a$. As $a > 0$ can be arbitrarily small, $\zeta(s)$ is analytic on Re(s) > 1.'

Common misconceptions

• Thinking that pointwise convergence of analytic functions is sufficient to guarantee the limit is analytic.
• Assuming that the Uniform Limit Theorem applies directly to the open region $\operatorname{Re}(s)>1$ without first restricting to closed sub-regions $\operatorname{Re}(s)\ge 1+a$.

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