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

TL;DR

This paper introduces a renormalization approach to evaluate the Riemann zeta function at -1, using difference methods and Cesaro means, providing new insights into the interpretation of the divergent series 1+2+3+... as -1/12.

Contribution

It presents novel difference-based and Cesaro mean techniques to extend the zeta function and interpret divergent series at -1, offering alternative perspectives to traditional methods.

Findings

Cesaro means yield convergence to zeta(-1)

Difference methods relate divergent sums to -1/12

New interpretation of 1+2+3+... as -1/12

Abstract

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) $