extension of $\zeta(s)=\lim_{n\to\infty}\sum_{k=1}^n 1/k^s -\int_0^n 1/x^s$ I saw $\zeta(1/2)=\lim_{n\to\infty}\sum_{k=1}^n 1/\sqrt k-\int_0^n\ 1/\sqrt x\ dx$
here
and similarly
$\zeta(s)=\lim_{n\to\infty}\sum_{k=1}^n 1/k^s -\int_0^n 1/x^s$ for $0<s<1$.
Rewriting this as $\zeta(s) = \lim_{n\to\infty}\sum_{k=1}^n 1/k^s -\int_1^n 1/x^s + 1/(s-1)$, I observed it also holds for $s > 1$. (It just follows from $\zeta$'s definition.)
(For example,
$\zeta(1/3) = \lim_{n\to\infty}\sum_{k=1}^n 1/k^3 -\int_1^n 1/x^{1/3} - 3/2$,
$\zeta(3) = \lim_{n\to\infty}\sum_{k=1}^n 1/k^3 -\int_1^n 1/x^3 + 1/2$.)
For $s=1$, I can interpret it as $\lim_{s\to 1} \zeta(s)-1/(s-1)=\lim_{n\to\infty}\sum_{k=1}^n 1/k -\int_1^n 1/x$, which is the Euler's constant.
Now I wonder is there some interpretation to apply this for $s\leq 0$?
(Some modification is needed because for example
$\zeta(-1) =-1/12 \neq \lim_{n\to\infty}\sum_{k=1}^n k -\int_1^n x + 1/2  = \lim_{n\to\infty} n/2+1/2 $.
but I could not find any natural modification.)
 A: Suppose that $\Re s>1$. In terms of the Hurwitz zeta function (http://dlmf.nist.gov/25.11)
$$
\sum\limits_{k = 1}^n {\frac{1}{{k^s }}}  = \zeta (s) - \sum\limits_{k = n + 1}^\infty  {\frac{1}{{k^s }}}  = \zeta (s) + \frac{1}{{n^s }} - \sum\limits_{k = n}^\infty  {\frac{1}{{k^s }}}  = \zeta (s) + \frac{1}{{n^s }} - \zeta (s,n),
$$
i.e.,
$$
\zeta (s) = \sum\limits_{k = 1}^n {\frac{1}{{k^s }}}  - \frac{1}{{n^s }} + \zeta (s,n).
$$
By analytic continuation, this formula is valid for all complex $s\neq 1$. Now you can use, for example, http://dlmf.nist.gov/25.11.E28 to obtain a more explicit expression that is valid in half-planes of the form $\Re s>-(2N+1)$. You can also use the asymptotic expansion
$$
\zeta (s,n) \sim \frac{{n^{1 - s} }}{{s - 1}} + \frac{1}{{2n^s }} + \sum\limits_{m = 1}^\infty  {\frac{{B_{2m} }}{{(2m)!}}\frac{{\Gamma (s + 2m - 1)}}{{\Gamma (s)}}\frac{1}{{n^{s + 2m - 1} }}} 
$$
as $n\to +\infty$ with fixed $s \neq 1$ (http://dlmf.nist.gov/25.11.E43). Here $B_m$ denotes the Bernoulli numbers. In particular,
$$
\zeta (s) = \mathop {\lim }\limits_{n \to  + \infty } \left( {\sum\limits_{k = 1}^n {\frac{1}{{k^s }}}  + \frac{{n^{1 - s} }}{{s - 1}} - \frac{1}{{2n^s }} + \sum\limits_{m = 1}^N {\frac{{B_{2m} }}{{(2m)!}}\frac{{\Gamma (s + 2m - 1)}}{{\Gamma (s)}}\frac{1}{{n^{s + 2m - 1} }}} } \right)
$$
for all $s\neq 1$ and $N$ satisfying $\Re s >  - (2N + 1)$. Note that
$$
\frac{{n^{1 - s} }}{{s - 1}} =  - \int_1^n {\frac{{dx}}{{x^s }}}  + \frac{1}{{s - 1}} .
$$
