Research article

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

  • Received: 04 May 2018 Accepted: 05 June 2018 Published: 21 June 2018
  • MSC : 11M99, 40C99

  • 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)  $.

    Citation: Gunduz Caginalp. A renormalization approach to the Riemann zeta function at — 1, 1 + 2 + 3 + …~  — 1/12.[J]. AIMS Mathematics, 2018, 3(2): 316-321. doi: 10.3934/Math.2018.2.316

    Related Papers:

  • 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)  $.



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    [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.
    [5] A. Padilla, E. Copeland, Available from: https://www.youtube.com/watch?v=w-I6XTVZXww.
    [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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