Approximating the Riemann Zeta Function with Continued Fractions

number of terms n

3

selected point

2

lower limit x axis

3

2

upper limit x axis

6

2

π

6

∼

16

9

1

1+

1

41+

7

36(1+0)

Continued fractions provide a very effective toolset for approximating functions. Usually the continued fraction expansion of a function approximates the function better than its Taylor or Fourier series. This Demonstration shows the high quality of a continued fraction expansion to approximate the Riemann ζ function.

The selected point (red), for which the continued fraction expansion is performed, must be between the two limits on the

x

axis. The number of terms used in the continued fraction expansion is

n

.

This continued fraction expansion is only correct for real numbers greater than 2; it cannot be used for general complex numbers.

The formula of the continued fraction expansion in this Demonstration is truncated when it would be too big to display.