What are the best approximations in terms of elementary functions of one real variable for:

$$\zeta(s)=\sum_{n=1}^{\infty} \frac{1}{n^s},$$ for $Re(s)>1?$

There is not an elementary function that equals $\zeta(s)$ so which one does best?

  • 1
    $\begingroup$ It's a Riemann sum for the integral $\int_1^{\infty} \frac{1}{x^s}dx = \frac{1}{s-1}.$ By summation by parts and Euler MacLaurin Summation, you can get good approximations. See also any analytic number theory text. $\endgroup$ – Dzoooks May 7 '19 at 0:43
  • $\begingroup$ I mean a curve of best fit for the zeta function $\endgroup$ – Ultradark May 7 '19 at 1:00
  • $\begingroup$ Not sure what you mean, but the techniques I've mentioned do give you arbitrarily good approximations. $\endgroup$ – Dzoooks May 7 '19 at 1:01

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