TITLE: How to show $\zeta (1+\frac{1}{n})\sim n$
QUESTION [5 upvotes]: How to show $\zeta (1+\frac{1}{n})\sim n$ as $n\rightarrow \infty$ where $\zeta$ is the Riemann zeta function.
And can we say $\lceil \zeta (1+\frac{1}{n}) \rceil=n$ for any positive integer $n\geq 1$. How to prove it?

REPLY [2 votes]: According to this page
$$\zeta(s)=\frac{1}{s-1}+\sum_{n=0}^\infty \frac{(-1)^n}{n!} \gamma_n \; (s-1)^n$$
the terms $γ_n$ being Stieltjes constants. So, if $s=1+\frac 1n$ $$\zeta\big(1+\frac 1n\big)=n+\sum_{i=0}^\infty \frac{(-1)^i}{i!} \gamma_i \; \frac 1{n^i}=n+\gamma+\sum_{i=1}^\infty \frac{(-1)^i}{i!}  \; \frac{\gamma_i } {n^i}$$