A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12.
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 gr...
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| Format: | Article |
| Language: | English |
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AIMS Press
2018-06-01
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| Series: | AIMS Mathematics |
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| Online Access: | http://www.aimspress.com/article/10.3934/Math.2018.2.316/fulltext.html |
| _version_ | 1819179918367391744 |
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| author | Gunduz Caginalp |
| author_facet | Gunduz Caginalp |
| author_sort | Gunduz Caginalp |
| collection | DOAJ |
| description | 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) $. |
| first_indexed | 2024-12-22T22:06:05Z |
| format | Article |
| id | doaj.art-e07954cd2e0948f0a7b6042c44ceba43 |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-12-22T22:06:05Z |
| publishDate | 2018-06-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-e07954cd2e0948f0a7b6042c44ceba432022-12-21T18:10:59ZengAIMS PressAIMS Mathematics2473-69882018-06-013231632110.3934/Math.2018.2.316A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12.Gunduz Caginalp0Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260, USAA 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) $.http://www.aimspress.com/article/10.3934/Math.2018.2.316/fulltext.htmlRiemann zeta function| sum of natural numbers| $\zeta\left(-1\right)$| 1+2+3+...| Cesaro mean or sum |
| spellingShingle | Gunduz Caginalp A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12. AIMS Mathematics Riemann zeta function| sum of natural numbers| $\zeta\left(-1\right)$| 1+2+3+...| Cesaro mean or sum |
| title | A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12. |
| title_full | A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12. |
| title_fullStr | A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12. |
| title_full_unstemmed | A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12. |
| title_short | A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~ — 1/12. |
| title_sort | renormalization approach to the riemann zeta function at 1 1 2 3 nbsp 1 12 |
| topic | Riemann zeta function| sum of natural numbers| $\zeta\left(-1\right)$| 1+2+3+...| Cesaro mean or sum |
| url | http://www.aimspress.com/article/10.3934/Math.2018.2.316/fulltext.html |
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