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.

Details

This is a special case of a continued fraction expansion for the polylog function, as
Li
n
(1)
is
ζ(n)
.
With the definitions
(r)
A
m
(n)=det
1≤i,j≤m
and
a
n,1
=1,
a
n,2m
=-
(1)
A
m
(n)
(0)
A
m-1
(n)
(0)
A
m
(n)
(1)
A
m-1
(n)
,
a
n,2m+1
=-
(1)
A
m-1
(n)
(0)
A
m+1
(n)
(0)
A
m
(n)
(1)
A
m
(n)
,
the continued fraction approximation for the polylog function can be written as
-
Li
n
(-z)=
∞
Κ
k=1
a
n,k
z
1
.
In particular, for
z=1
,
ζ(n)=
1
(1-
1-n
2
)
∞
Κ
k=1
a
n,k
1
as well as
log2=
∞
Κ
k=1
a
1,k
1
.

External Links

Continued Fraction (Wolfram MathWorld)
Polylogarithm (Wolfram MathWorld)
Riemann Zeta Function (Wolfram MathWorld)

Permanent Citation

Andreas Lauschke
​
​"Approximating the Riemann Zeta Function with Continued Fractions"​
​http://demonstrations.wolfram.com/ApproximatingTheRiemannZetaFunctionWithContinuedFractions/​
​Wolfram Demonstrations Project​
​Published: March 7, 2011