AIMS Mathematics, 2018, 3(2): 316-321. doi: 10.3934/Math.2018.2.316

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

Mathematics Department, University of Pittsburgh, Pittsburgh, PA 15260, USA

## Abstract    Full Text(HTML)    Figure/Table

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$\zeta\left( -1\right)$.
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# References

1. E. C. Titchmarsh, D. R. Heath-Brown, The theory of the Riemann zeta-function, Oxford University Press, 1986.

2. K. G.Wilson, J. B. Kogut, The renormalization group and the $\epsilon$ expansion, Phys. Rep., 12 (1974), 75–200.

3. R. J. Creswick, C. P. Poole and H. A. Farach, Introduction to renormalization group methods in physics, 1992.

4. J. G. Polchinski, String Theory, Volume I, An Introduction to the Bosonic String, Cambridge University Press, 1998.

6. E. M. Stein, R. Shakarchi, Fourier Analysis an introduction, Princeton University Press, 2003.

7. P. Billingsley, Probability and Measure, Wiley: Anniversary Edition, 2012.

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