It is well known that the infinite sum:
$$\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$$
only converges for $\Re(s)>1$.
The Dirichlet 'alternating' sum:
$$\displaystyle \zeta(s) = \frac{1}{1-2^{1-s}}\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}$$
allows for analytic continuation towards $\Re(s)>0$, but also does introduce additional zeros.
There are also nested sums see formulas 21 and 22 here that fully extend $\zeta(s)$ towards $s \in \mathbb{C}/1$.
However, I found this surprisingly simple infinite sum that Maple evaluates correctly for $\Re(s) >-1$:
$$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(s+1+ \sum _{n=1}^{\infty } \left( {\frac {s-1-2\,n}{{n}^{s}}}+{\frac {s+1+2\,n}{\left( n+1 \right) ^{s}}}\right) \right)$$
Following Peter's suggestion below, it can be easily shown that by splitting, re-indexing via $n \mapsto n-1$ and then recombining the sums, that for $\Re(s) > 1$ the formula correctly reduces to $\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$.
For the analytically continued domain $-1<\Re(s)<1$, the operations of re-indexing and recombining the split sums are no longer allowed, since the sums are now diverging.
I played a bit with the individual terms in the infinite sum and it is easy to see that for $s=0$ and $s=1$ all terms will be zero. Therefore $\zeta(0)$ will be fully determined by $\frac{s+1}{2\,(s-1)} = \frac{1}{-2}=-\frac12$. When $s\rightarrow 1$, there will be a pole induced by $\frac{s+1}{2(s-1)}$, however the pole $\frac{1}{2(s-1)}$ will be offset by the infinite sum approaching $0$ and converges to the nice result $\gamma -\frac12$ with $\gamma$ being the Euler-Mascheroni constant.
Did not have much time to go deeper, but did find a nice result for $s=2$ (n=1..7):
$s=2 \rightarrow \frac{1 }{4}+ \frac{1 }{36}+\frac{1}{144}+\frac{1}{400}+\frac{1}{900}+\frac{1}{1764}+\frac{1}{3136}+\dots$ = sum of 1/A035287
For $s=3$ the following pattern emerges, however I did not find a good link to Sloane's.
$s=3 \rightarrow \frac{3 }{4}+\frac{5 }{108}+\frac{7}{864}+\frac{9}{4000}+\frac{11}{13500}+
\frac{13}{37044}+\frac{15}{87808}+\dots$
Questions:
(1) Has anyone seen this formula before? If so, I'd be grateful for a reference.
(2) To avoid this is just Maple playing a trick on me here, I am curious to learn whether the results can be reproduced in other CAS. (EDIT: now confirmed in Maple, Sage, Pari/GP and Mathematica that the formula works correctly, although numerical discrepancies are found for higher values $> 10^8$ of $\Im(s)$).
Final addition:
Paul Garrett´s reference below gives a good method to construe the formula above. However, I realized myself that I posted the formula on MO without any explanation on how I did find it in the first place. So, here it is to make the question complete before it vanishes into MO´s history:
Assume $s \in \mathbb{C}$, $\Re(s) \ge 0$ and take the well known expression:
$$\zeta(s) = \dfrac{s}{s-1} – \frac12+s \int_1^\infty \frac{1/2-\{x\}}{x^{s+1}}\,\mathrm{d}x$$
and substitute the fractional part of $\{x\}$ by a closed form (see here):
$$\displaystyle \{x\} = x – \lfloor x \rfloor = -\frac{\arctan\left(\cot\left(\pi x\right)\right)}{\pi} + \frac{1}{2}$$
which gives:
$$\displaystyle Z(s) = \dfrac{s}{s-1} – \frac12 +s \int_1^\infty \frac{\arctan\left(\cot\left(\pi x\right)\right)}{\pi x^{s+1}}\mathrm{d}x $$
For all rational fractions $-1<\frac{k}{n}<1$, Maple consistently returns closed forms with an infinite sum e.g.:
$$Z \left(\frac23\right) =\displaystyle 1-\frac{11}{4}\,\sqrt [3]{2}+\frac{2}{3}\,\sum _{{\it n}=2}^{\infty }-\frac{3}{4}\,{\frac {- \left( {\it n}+1 \right) ^{2/3}-6\, \left( {\it n}+1 \right) ^ {2/3}{\it n}+5\,{{\it n}}^{2/3}+6\,{{\it n}}^{5/3}}{{{\it n}}^{2/3} \left( {\it n}+1 \right) ^{2/3}}}$$
After experimenting with a few rationals, I saw a clear pattern emerge that I then could easily generalize into the formula above. To my surprise it also worked for irrational values and the domain $\Re(s) \ge 1$. Not sure whether Maple uses a similar technique as Paul Garrett´s suggestion below to transform the integral of the fractional parts into these infinite sums.
Best Answer
This can be construed as an application of Euler-Maclaurin summation, as carried out to an arbitrary number of stages in Appendix B of Montgomery-Vaughan's "Multiplicative Number Theory I: Classical Theory".
The general idea is to subtract the integral corresponding to a sum, breaking the integral into integrals over subintervals, as in the well-known idea $$ \zeta(s) - {1\over s-1} \;=\; \sum_n \Big({1\over n^s} - \int_n^{n+1}{dx\over x^s}\Big) \;=\; \sum_n \Big({1\over n^s} - {n\over (s-1)n^s} + {n+1\over (s-1)(n+1)^s}\Big) $$ Elementary estimates show that the new sum is absolutely and uniformly-on-compacts convergent on $\Re(s)>0$. One can repeat the approximation of sum by integral as many times as desired, although organization becomes an issue. The sum in the question is equivalent to application of this idea one more stage. Montgomery-Vaughan show how to systematize this and express an arbitrary (finite) number of iterates in terms of Bernoulli polynomials/numbers.