Zeta?

Relevant wiki: Riemann zeta function

Whittaker and Watson (1990, p. 271) (eqn. 15) showed that for complex number $s$ , where $\Re[s]>1$ , as $s \to 1$ : $\lim_{s \to 1} \left[\zeta(s)-\dfrac 1{s-1} \right]=\gamma$ where $\gamma$ is the Euler-Mascheroni constant .

Therefore, we have:

$\begin{aligned} \lim_{x \to 1} \left[\zeta(x)-\dfrac 1{x-1} \right] & = \gamma \\ \lim_{x \to 1^+} \left[(x-1) \zeta(x)-\dfrac {x-1}{x-1} \right] & = \lim_{x \to 1^+} (x-1)\gamma \\ \lim_{x \to 1^+} (x-1) \zeta(x)-1 & = 0 \\ \implies \lim_{x \to 1^+} (x-1) \zeta(x) & = \boxed{1} \end{aligned}$

We define $\displaystyle \eta(s) = (1-2^{1-s})\zeta(s)$ where $s>1$ , where $\displaystyle \eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^s}$ .

$\displaystyle \lim_{x\to1^+}(x-1)\zeta(x) = \displaystyle \lim_{x\to1^+} \frac{\eta(x)}{\frac{1-2^{1-x}}{x-1}} = \displaystyle \lim_{x\to1^+} \frac{\eta(x)}{ln2} = \displaystyle \frac{\eta(1)}{ln2}=1$