In his short note on Euler and the prime harmonic series, Professor of Mathematics Paul Pollack, makes the following intermediate statement about the Riemann zeta function $\zeta(s)$ for $s>1$, without explanation:
$$ 1<s\cdot\zeta(s+1) < \frac{3}{2} $$
Question: Where does that $\frac{3}{2}$ comes from?
My thinking ...
He says it follows from a lemma which in his note is the well known inequality:
$$ \frac{1}{s-1} <\zeta(s)< \frac{s}{s-1} $$
.. which he expresses as
$$1<(s-1)\cdot \zeta(s)<s$$
Substituting $(s+1)$ for $s$ would lead to the following inequality:
$$1 < s \cdot \zeta(s+1) < s+1$$
In the document linked, Lemma 3 provides the general result that $$1<(s-1)\zeta(s)<s$$ for every $s$, but, the Lemma 4 statement which contains the $\frac 32$ restricts $s$ to $(0,\frac12)$. Indeed, in this case, we have that for each $s\in(0,\frac12)$: $$1<s\zeta(s+1)<\frac12\zeta \left(\frac32\right)<\frac32$$