A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~  — 1/12.

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)  $.