# Elementary approximations to $\zeta(s)?$

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?

• 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. – Dzoooks May 7 '19 at 0:43
• I mean a curve of best fit for the zeta function – Ultradark May 7 '19 at 1:00
• Not sure what you mean, but the techniques I've mentioned do give you arbitrarily good approximations. – Dzoooks May 7 '19 at 1:01