The celebrated Riemann $\zeta(s)$ has the following form, for all $s \in \mathbb{C}$ with real part greater than $1$

$$

\begin{equation}

\zeta(s) = \sum_{n \geq 1} n^{-s}

\end{equation}

$$

Actually, $\zeta(s)$ can be analytically continued over all $\mathbb{C} - \{1\}$, with a unique pole at $1$.

To define Chaitin $\Omega$, we first have to fix some universal prefix-free partial recursive function $U : \{0,1\}^* \rightarrow \{0,1\}^*$. This simply means that we fix some computer whose legal programs, encoded as binary strings, have some sort of "end of file" marker: no legal program is the prefix of another legal program. In addition, this computer is universal, i.e., is able to simulate every other computer. Denoting by $Dom(U)$ the set of legal programs of $U$, the Chaitin's number is

$$

\Omega = \sum_{p \in Dom(U)} 2^{-|p|}

$$

where $|p|$ denotes the length of the program $p$. This number can be interpreted as a probability. Indeed, if we flip infinitely many times a fair coin, then $2^{-|p|}$ is the probability that the prefix of length $|p|$ in the generated infinite binary sequence is equal to $p$. Since $U$ is prefix-free, this also represents the probability that $U$ will execute the program $p$ and halt. Hence, $\Omega$ is the probability that $U$ halts when given the generated infinite binary sequence. It can be shown that the binary expansion of $\Omega$ is a Martin-Löf random sequence.

In [1], Tadaki generalizes Chaitin $\Omega$, and define, for all $0 < D < 1$

$$

\begin{equation}

\Omega(D) = \sum_{p \in Dom(U)} 2^{-\frac{|p|}{D}}

\end{equation}

$$

Now, I guess everyone has caught it, but let's put bluntly the (not so) funny part. For all (real) $s > 1$

$$

\begin{align}

\zeta(s) &= \sum_{n \geq 1} \left( 2^{\log n}\right)^{-s} \\

\Omega(s^{-1}) &= \sum_{p \in Dom(U)} \left( 2^{|p|} \right)^{-s}

\end{align}

$$

Of course, these two quantities are not equal: $\Omega(s^{-1})$ is a sub-sum of $\zeta(s)$ which does not pick any term with $n$ not a power of $2$. We can make them look even more similar by introducing, for every $n$, the function $m(l) = |\{ p \in Dom(U),~ |p| = l\}|$ counting the number of legal programs of length $l$.

$$

\begin{align}

\zeta(s) &= \sum_{n \geq 1} n^{-s} \\

\Omega(s^{-1}) &= \sum_{n \geq 1} m(\log n) \cdot n^{-s}

\end{align}

$$

The definition of $\Omega(s^{-1})$ really looks like a form of $\zeta$ regularization of the counting function $m(\log n)$. Actually, we could even define, for any universal (not necessarily prefix-free) function $C$, with $m_C(\log n)$ counting the number of legal programs of $C$ with length $\log n$

$$

\begin{equation}

\Omega_C(s^{-1}) = \sum_{n \geq 1} m_C(\log n) \cdot n^{-s}

\end{equation}

$$

provided that the real part of $s$ is sufficiently large. In particular, because the domain of $C$ is not necessarily prefix-free, the function may diverge when $s \rightarrow 1$.

*. Does such a regularization give more insight on random strings ?*

**Question**
pb

EDIT: It turns out that Tadaki has already proposed an analogy [2] between $\Omega(s^{-1})$ and partition functions in statistical physics. Since $\zeta$ naturally arises in statistical physics, the above similarity between $\Omega(s^{-1})$ and $\zeta(s)$ is not a surprise.

EDIT: Truly, there is something going on here. I found this paper [3] by Calude, introducing zeta numbers for Turing machines.

[1] Tadaki, Kohtaro.

[2] Tadaki, Kohtaro.

[3] Cristian S. Calude, Michael A. Stay, Natural halting probabilities, partial randomness, and zeta functions, Information and Computation, Volume 204, Issue 11, November 2006, Pages 1718-1739, ISSN 0890-5401, http://dx.doi.org/10.1016/j.ic.2006.07.003.

*A generalization of Chaitin's halting probability $\Omega$ and halting self-similar sets*. Hokkaido Mathematical Journal 31 (2002), no. 1, 219--253. doi:10.14492/hokmj/1350911778. http://projecteuclid.org/euclid.hokmj/1350911778[2] Tadaki, Kohtaro.

*A statistical mechanical interpretation of algorithmic information theory*. http://arxiv.org/abs/0801.4194[3] Cristian S. Calude, Michael A. Stay, Natural halting probabilities, partial randomness, and zeta functions, Information and Computation, Volume 204, Issue 11, November 2006, Pages 1718-1739, ISSN 0890-5401, http://dx.doi.org/10.1016/j.ic.2006.07.003.

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