I do not mean the continued fraction representation of zeta function; I mean the function which has the form:


For some values, we know the function exactly:

$$\begin{align*} f(0)&=\phi-1=0.61803398875\ldots\\ f(1)&=\frac{I_1(2)}{I_0(2)}=0.69777465796\ldots\\ f(-1)&=\frac{\pi}{2}-1=0.5707963268\ldots \end{align*}$$

And obviously, for big positive $s$, it approaches $1$.

$$f(s \rightarrow +\infty) \rightarrow 1$$

Using the convergence criterion for simple continued fractions, we can conclude that $f(s<-1)$ doesn't converge, which is confirmed by the behavior of its approximants.

Here are the first few approximations, based on truncating the continued fraction:

$$\begin{align*} f_1(s)&=1\\ f_2(s)&=\frac{1}{1+2^{-s}}\\ f_3(s)&=\frac{2^s+3^{-s}}{1+2^s+3^{-s}}\\ f_4(s)&=\frac{1+2^s(3^s+4^{-s})}{1+(1+2^s)(3^s+4^{-s})} \end{align*}$$

Defining (thanks to vrugtehagel for his helpful comment):

$$f(s)=\lim_{n \rightarrow \infty} f_n(s)$$

CF zeta approx

How would you analyze this function? Can it be turned into a series? Which of its properties can be found theoretically? Maybe it was already studied, then I would like some reference.


Using recurrence relations for continued fractions, I was able to get approximants for $f(s)$ of any order. The branching point near $s=-1$ is easy to see.

some approximants

  • $\begingroup$ Note that the function is actually a limit, since infinite fractions don't really make sense $\endgroup$ – vrugtehagel Feb 17 '16 at 22:09
  • 1
    $\begingroup$ $f(2)$ is oeis.org/A073824, but sadly, the OEIS entry doesn't say much else about it. $\endgroup$ – Chris Culter Mar 29 '16 at 18:26
  • $\begingroup$ Also, the sequence of digits given by $f\left(\frac{1}{2}\right) = 0.65266376...$ is not currently in the OEIS, and the original Inverse Symbolic Calculator (see wayback.cecm.sfu.ca/projects/ISC/ISCmain.html) does not 'recognize' this number. $\endgroup$ – John M. Campbell Mar 30 '16 at 18:24

(This is a comment that got too long for the comment box.)

For fun, I decided to implement this function for complex arguments in Mathematica, and plot its real and imaginary parts. Here's the picture I got:

real and imaginary parts for a "Dirichlet continued fraction"

That pole fence jives with the OP's observation that the function is only sensible for $\Re s > -1$.

Can it be turned into a series?

Certainly, you can at least build an Euler-Minding series out of this CF:

$$\sum_{k=0}^\infty\frac{(-1)^k}{Q_k Q_{k+1}}$$

where $Q_k$ is the denominator of the $k$-th convergent, and satisfies the recurrence $Q_k=k^s Q_{k-1}+Q_{k-2}$ with $Q_0=Q_1=1$. I don't know if any other interesting series can be easily built, since I am unable to find a usable closed form for $Q_k$.

For those who want to try it out in Mathematica, I implemented the Lentz-Thompson-Barnett recurrence (also mentioned here) like so:

w[z_?InexactNumberQ] := 
  Module[{prec = Internal`PrecAccur[z], b, c, d, f, h, k},
         f = c = 1; d = 0; k = 2;
         While[b = k^z;
               d = 1/(b + d); c = b + 1/c;
               f *= (h = c d); k++;
               Abs[h - 1] > 10^-prec && k < 1000];

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.