= Zeta bounds from rounded Pareto variables
{title2=$1/(s-1)\le\zeta(s)\le s/(s-1)$}
Let $X$ have <survival function> $\mathbb P(X\ge x)=x^{-s}$ for $x\ge1$, with $s>1$. Its <expected value> is $s/(s-1)$. The <tail-sum formula for expectation> gives
$$
\mathbb E[\lfloor X\rfloor]=\sum_{k\ge1}k^{-s}=\zeta(s).
$$
The <absolutely continuous probability distribution> gives zero mass to the <integers>, so $\lceil X\rceil=\lfloor X\rfloor+1$ almost surely and $\mathbb E[\lceil X\rceil]=\zeta(s)+1$. Taking expectations of $\lfloor X\rfloor\le X\le\lceil X\rceil$ gives $\zeta(s)\le s/(s-1)\le\zeta(s)+1$, which rearranges to the displayed bounds on the <Riemann zeta function>. Finiteness is justified by $\lfloor X\rfloor\le X$ and $\lceil X\rceil\le X+1$.
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