A question from Titchmarsh's Riemann Zeta Function textbook.

Question

I have one query, concerning the newest edition of this monograph.

1. At page 7, section 1.2, at the bottom of the page, it's written that: " It is easily seen that $\zeta(s)=2$ for $s=\alpha$, where $\alpha$ is a real number greater than 1..."

I am quite sure that for $\alpha=2$ which is real and greater than 1, $ \zeta(s)=\frac{\pi^2}{6} \neq 2$, what do you think did Titchamarsh meant here?!

Thanks in advance.

Answer 1

I will take a stab at this. I think he is saying that for some real number $s > 1$, $\zeta(s) = 2.$ We know that on the interval $(1,\infty)$ the $\zeta$ function is continuous and monotonically decreasing. Since $\zeta(t) \to \infty$ as $t \downarrow 1$ and $\zeta(2) = \pi^2/6 < 2$, there must be some $s \in(1,2)$ with $\zeta(s) = 2.$

His writing somewhat obscures the existential nature of the statement he is making.