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

Abstract

A scaling and renormalization approach to the Riemann zetafunction, $\zeta$, 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 meantwice, i.e., $\left( C,2\right) $, yields convergence to the appropriatevalue. For values of $z$ for which the zeta function is represented by a\textit{convergent} infinite sum, the double Cesaro mean also yields$\zeta\left( z\right) ,$ suggesting that this could be used as analternative method for extension from the convergent region of $z.$ In thesecond approach, the difference $U_{n}-k^{2}\bar{U}_{n/k}$ between $U_{n}$ anda particular average, $\bar{U}_{n/k}$, involving terms up to $k<n$ and scaledby $k^{2}$ 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) $.