  A scaling and renormalization approach to the Riemann zeta function, $\zeta$,
evaluated at $-1$ is presented in two ways. In the first, one takes the
difference between
  $U_{n}:=\sum_{q=1}^{n}q$ and $4U_{\left\lfloor \frac{n}{2}\right\rfloor }$
where $\left\lfloor \frac{n}{2}\right\rfloor $ is the greatest integer
function. Using the Cesaro mean twice, i.e., $\left( C,2\right) $, yields
convergence to the appropriate value. For values of $z$ for which the zeta
function is represented by a convergent infinite sum, the double Cesaro mean
also yields $\zeta\left( z\right) ,$ suggesting that this could be used as an
alternative method for extension from the convergent region of $z.$ In the
second approach, the difference $U_{n}-k^{2}\bar{U}_{n/k}$ between $U_{n}$ and
a particular average, $\bar{U}_{n/k}$, involving terms up to $k<n$ and scaled
by $k^{2}$ is shown to equal exactly $-\frac{1}{12}\left( 1-k^{2}\right) $ for
all $k<n$. This leads to another perspective for interpreting $\zeta\left(
-1\right) $
