English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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) $.

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