Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Riemann's zeta function is a function with many faces, I mean representations. I recently derived this one, bellow, as a continued fraction over prime numbers.

$$ \zeta(s)=1 +\cfrac{\frac{1}{2^{s}}}{1-\frac{1}{2^{s}} -\cfrac{\frac{2^{s}-1}{3^{s}}}{1+\frac{2^{s}-1}{3^{s}} -\cfrac{\frac{3^{s}-1}{5^{s}}}{1+\frac{3^{s}-1}{5^{s}} -\cfrac{\frac{5^{s}-1}{7^{s}}}{1+\frac{5^{s}-1}{7^{s}} -\cfrac{\frac{7^{s}-1}{11^{s}}}{1+\frac{7^{s}-1}{11^{s}} -\ddots}}}}} $$

... and I'd like to know if this is known in the literature and if so I'd appreciate to have references about it.


share|cite|improve this question
"Found" in the sense of derived it yourself, or found it in a written source somewhere? If the latter, you might include that source here. – Jack M Apr 8 '14 at 11:50
@Jack M, I derived it myself. – Neves Apr 8 '14 at 11:54
Please show your derivation. Thank you. – marty cohen Apr 11 '14 at 2:46
@marty, ok, I'll do that. – Neves Apr 11 '14 at 6:23
@FredKline, Thanks, but I already knew this one... and none of the others is "over primes". – Neves Apr 12 '14 at 6:14
up vote 6 down vote accepted

The continued fraction representation above had its origins on another problem I was working on sometime ago.

It's based on a very simple way of looking at the Euler's product representation of $\frac{1}{\zeta(s)}$. Interestingly it applies to every infinite product.

And this is as follows

$$ \frac{1}{\zeta(s)}=\left(1-\frac{1}{2^s}\right)-\left(1-\frac{1}{2^s}\right)\frac{1}{3^s}-\left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{3^s}\right)\frac{1}{5^s}-\cdots $$

From here its easy to derive the above continued fraction using Euler's continued fraction formula.

And thats it, It's nice and eventually a new thing.


Just to make it clear, note that $$ \begin{align*} \frac{1}{\zeta(s)}&=\left(1-\frac{1}{2^s}\right)\left[\left(1-\frac{1}{3^s}\right)-\left(1-\frac{1}{3^s}\right)\frac{1}{5^s}-\left(1-\frac{1}{3^s}\right)\left(1-\frac{1}{5^s}\right)\frac{1}{7^s}-\cdots\right]\\ &=\left(1-\frac{1}{2^s}\right)\left(1-\frac{1}{3^s}\right)\left[\left(1-\frac{1}{5^s}\right)-\left(1-\frac{1}{5^s}\right)\frac{1}{7^s}-\cdots\right]\\ &\vdots\\ &=\prod_{p\in\mathbb{P}}\left(1-\frac{1}{p^{s}}\right) \end{align*} $$ where $\mathbb{P}$ is the set of the prime numbers.


To derive the continued faction just put $\frac{1}{\zeta(s)}$ in the form $$ \frac{1}{\zeta(s)}=1-\frac{1}{2^s}\left(1+\frac{2^s-1}{3^s}\left(1+\frac{3^s-1}{5^s}\left(1+\frac{5^s-1}{7^s}\left(1+\frac{7^s-1}{11^s}\left(1+\ddots\right ) \right ) \right ) \right ) \right) $$ and then just apply the Euler continued fraction formula. So we can write this as $$ \frac{1}{\zeta(s)}=1-\frac{1}{2^s}-\frac{1}{2^s}\frac{2^s-1}{3^s}-\frac{1}{2^s}\frac{2^s-1}{3^s}\frac{3^s-1}{5^s}-\frac{1}{2^s}\frac{2^s-1}{3^s}\frac{3^s-1}{5^s}\frac{5^s-1}{7^s}-\cdots $$ Now, let $a_1=-\frac{1}{2^s};a_2=\frac{2^s-1}{3^s};a_3=\frac{3^s-1}{5^s};a_4=\frac{5^s-1}{7^s}\cdots$ and we'll get the Euler continued fraction formula.

share|cite|improve this answer
Found this: – Fred Kline Apr 15 '14 at 3:50
Can you expand on the last step.? I didn't get it. – user230452 Mar 29 at 23:01
@user230452, I added some clarification. – Neves Mar 30 at 21:08

By using Mathematica to simplify the first 7 primes, we get: $$\frac{510510^s}{\left(2^s-1\right) \left(3^s-1\right) \left(5^s-1\right) \left(7^s-1\right) \left(11^s-1\right) \left(13^s-1\right) \left(17^s-1\right)},$$ which is equivalent to: $$\prod _{p\text{ prime}} \frac{p^s}{p^s-1} = \zeta(s).$$ Product does not converge when $s=1.$

Scroll down to Euler product formula (2nd paragraph).
When $s=1\text{, }\frac{1}{1-\frac{1}{p^s}}$ simplifies to $\frac{p}{p-1}.$ When $s>1,$ there is no simplification.

Neves's formula puts the exponents back onto the primes when it is simplified, $\frac{p^s}{p^s-1}.$

share|cite|improve this answer
Thats $\zeta(s)$ in the Euler's product form... – Neves Apr 11 '14 at 6:18
@Neves, we crossed paths. My last edit explains. – Fred Kline Apr 11 '14 at 7:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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