Critical Dimension of Bosonic Strings and Regularization of $sum_{n=1}^infty n$

If $D$ is critical dimension of Bosonic strings, a particular derivation goes like the following, where we arrive finally at

$$\frac{D-2}{2}\sum_{n=1}^\infty n + 1 = 0.$$ Now mathematically this is clearly a divergent series, but using zeta function regularization here we are taking $$\sum_{n=1}^\infty n = \zeta(-1) = -\frac{1}{12}.$$ And obtain $D = 26$ where $\zeta$ is the analytic continuation of the zeta function we know. But it makes no sense in putting $s = -1$ in the formulae $$\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.$$ As the above is only valid for $Re(s) > 1$. So what is going on in here? Can anyone give me a reasonable explanation about obtaining $-1/12$?

Answer

I know some derivations in which one can track the emergence of the concrete value, without having to buy that the second order contribution in the Euler-MacLaurin formula (see other answer) is $-\frac{1}{2!}$ times the second Bernoulli number $B_2$.

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The limit $\lim_{z\to 1}$ of the sum

$0+1\,z^1+2\,z^2+3\,z^3+\dots$

diverges, because of the pole in

$\sum_{k=0}^\infty k\,z^k=z\frac{{\mathrm d}}{{\mathrm d}z}\sum_{k=0}^\infty z^k=z\frac{{\mathrm d}}{{\mathrm d}z}\frac{1}{1-z}=\frac{z}{(z-1)^2}, \hspace{1cm} z\in(0,1)$

We are instead going to consider the sum of smooth deviations of the above, using the local mean

$\langle f(k)\rangle:=\int_{k}^{k+1}f(k')\,{\mathrm d}k'$.

for which $\langle k\,z^k\rangle=z\frac{{\mathrm d}}{{\mathrm d}z}\langle z^k\rangle=z\frac{{\mathrm d}}{{\mathrm d}z}\langle {\mathrm e}^{k \log(z)}\rangle=z\frac{{\mathrm d}}{{\mathrm d}z}\frac{z^{k'}}{\log(z)}\left|_{k}^{k+1}\right.$.

Because of canceling upper and lower bounds, the sum $\sum_{k=0}^n\langle k\,z^k\rangle$ is $\frac{z^0}{\log(z)^2}$ plus terms suppressed by $z^n$. Finally, using the expansion

$\dfrac{1}{\left(\log(1+r)\,/\,r\right)^2}=\dfrac{1}{1-r+\left(1-\frac{1}{1!\,2!\,3!}\right)r^2+{\mathrm{O}}(r^3)}=1+r+\dfrac{1}{1!\,2!\,3!}r^2+{\mathrm{O}}(r^3),$

we find

$\sum_{k=0}^\infty \left(k\,z^k-\langle k\,z^k\rangle\right)=\dfrac{z}{(z-1)^2}-\dfrac{1}{\log(z)^2}=-\dfrac{1}{12}+{\mathcal O}\left((z-1)^1\right).$

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The picture shows the two functions $\dfrac{z}{(z-1)^2}$ and $\dfrac{1}{\log(z)^2}$, as well as their difference (blue, red, yellow). While the functions themselves clearly have a pole at $z=1$, their difference converges against

$$-\frac{1}{1!\,2!\,3!}=-\frac{1}{12}=-0.08{\dot 3}.$$