# Prime Zeta Function

The **prime zeta function** is an expression similar to the Riemann zeta function. It has interesting properties that are related to the properties of the Riemann zeta function, as well as a connection to Artin's conjecture about primitive roots.

The prime zeta function \(\zeta_{\mathbb{P}}(s)\), where \( s \) is a complex number, is defined by the series \[ \zeta_{\mathbb{P}}(s) = \sum_{p \in \mathbb{P}} \dfrac{1}{p^{s}} \] where \(\mathbb{P}\) is the set of prime numbers.

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## Divergence of \(\zeta_{\mathbb{P}}(1)\)

The value of \( \zeta_{\mathbb{P}}(1) \) is the sum of the reciprocals of the primes. This series diverges, but very slowly:
\[
\sum_{\substack{p \in \mathbb{P} \\ p < n}} \frac1{p} - \log\big(\log(n)\big) \to M\ \text{ as }\ n\to \infty,
\]
where \( M = 0.261497\ldots\) is a constant (called the **Meissel-Mertens constant**). This is reminiscent of the definition of the Euler-Mascheroni constant \( \gamma\).

Euler asserted that the sum of the reciprocals of the primes diverged, and derived a correct estimate for it, but his proof involved manipulations of divergent infinite series and products (for instance, the Euler product expansion for \( \zeta(s) \) evaluated at \( s=1 \)). As is often the case with Euler's arguments, it can be made rigorous with some extra work.

## Expression in terms of the Riemann Zeta Function

The Euler product for \( \zeta(s) \) \(\big(\)for \(\text{Re}(s)>1\big)\) is \[ \zeta(s) = \prod_{p \text{ prime}} \left( 1-\frac1{p^s} \right)^{-1} \] and taking the natural log of both sides gives \[ \begin{align} \log\big(\zeta(s)\big) &= - \sum_{p \text{ prime}} \log\left( 1-\frac1{p^s} \right) \\ &= \sum_{p \text{ prime}} \sum_{k=1}^{\infty} \frac{\hspace{1.5mm} \frac{1}{p^{ks}}\hspace{1.5mm} }{k}, \end{align} \] using the Maclaurin series \( -\log(1-x) = x+\frac{x^2}2 + \frac{x^3}3 + \cdots. \)

Now switching the two sums gives \[ \begin{align} \log\big(\zeta(s)\big) &= \sum_{k=1}^{\infty} \sum_{p \text{ prime}} \frac{\hspace{1.5mm} \frac{1}{p^{ks}}\hspace{1.5mm} }{k} \\ &= \sum_{k=1}^{\infty} \frac1{k} \sum_{p \text{ prime}} \frac1{p^{ks}} \\ &= \sum_{k=1}^{\infty} \frac{\zeta_{\mathbb{P}}(ks)}{k}. \end{align} \] A generalization of Möbius inversion says that \[ f(x) = \sum_{k=1}^{\infty} \frac{g(kx)}{k} \Leftrightarrow g(x) = \sum_{k=1}^{\infty} \mu(k)\frac{f(kx)}{k}, \] where \( \mu \) is the Möbius function, as long as the sums are absolutely convergent (the proof is straightforward). Applying this gives \[ \zeta_{\mathbb{P}}(s) = \sum_{k=1}^{\infty} \mu(k) \frac{\log\big(\zeta(ks)\big)}{k}. \] Note that this gives an idea of why \( \zeta_{\mathbb{P}}(1) \) diverges at the same speed as \( \log\big(\log(n)\big) \), since the \( k=1 \) term is the only undefined term at \( s= 1, \) and \( \zeta(1) \) diverges like \( \log(n) \). In fact, expanding near \( s=1 \) gives \((\)for \( x > 0) \) \[ \zeta_{\mathbb{P}}(1+x) = -\log(x)+(M-\gamma) + O(x), \] where \( \gamma \) is the Euler-Mascheroni constant. The \( O(x) \) terms go to \( 0 \) as \( x \to 0^+ .\)

## Connection with the Twin Prime Constant

The twin prime conjecture is that there are infinitely many primes \( p \) such that \( p+2 \) is also prime. While this is still open, heuristics suggest that it is true and that in fact the function \(\pi_2(x) \) that counts twin primes \( \le x \) satisfies
\[
\pi_2(x) \sim 2\Pi_2 \int_2^x \frac{dx}{\big(\log(x)\big)^2},
\]
where \( \Pi_2 =0.66016\ldots\) is the **twin prime constant**
\[
\sum_{p \text{ odd prime}} \left( 1-\frac1{(p-1)^2}\right).
\]

A computation involving Taylor expansions of logarithms (similar to the above one) shows that the constant \( \Pi_2 \) is related to the values of \( P(s) \) as follows: \[ \log(\Pi_2) = - \sum_{k=2}^{\infty} \frac{2^k-2}{k} \big(\zeta_{\mathbb{P}}(k)-2^{-k}\big). \] The point is that the prime zeta function comes up in evaluating constants involving products over all primes.

## Connection with Artin's Conjecture

**Artin's conjecture** states that if \( a \) is an integer that is not a perfect square or \( -1 \), then it is a primitive root modulo \( p \) for infinitely many \( p. \) As usual, there is a heuristic estimate for the probability that \( a \) is a primitive root mod a given prime \( p. \) If \( a \) is squarefree and not congruent to \( 1 \) mod \( 4\), this probability is conjectured to be **Artin's constant**
\[
C = \prod_{p \text{ prime}} \left( 1-\frac1{p(p-1)} \right) = 0.37395\ldots.
\]
(For other values of \( a \), there is also a conjectural probability that is a rational multiple of Artin's constant.)

A calculation involving taking logs and expanding gives \[ \log(C) = \sum_{n=2}^{\infty} (1-L_n) \frac{\zeta_{\mathbb{P}}(n)}{n}, \] where \( L_n \) is the \( n^\text{th}\) Lucas number \(\big(\)defined by \( L_1 = 1, L_2 = 3, L_n = L_{n-1}+L_{n-2}\big). \)

## Generalization

An interesting generalization can be made by summing over inverse of positive integers raised to power \(s\) which are a product of \(k\) primes (not necessarily distinct).

The \(k\) prime zeta function \(\zeta_{\mathbb{P}}(k,s)\), where \( s \) is a complex number and \(k\) is a non-negative integer, is defined by the series \[ \zeta_{\mathbb{P}}(k,s) = \sum_{n : \Omega(n) = k} \dfrac{1}{n^{s}}, \] where \(\Omega\) denotes the number of prime factors.

## Relation with other Zeta Functions

Try proving these interesting identities involving generalized prime zeta function:

\[\begin{align} \sum_{k=0}^{\infty}\zeta_{\mathbb{P}}(k,s) &= \zeta (s) \\\\ \lim_{s \to 1} (s-1) \zeta_{\mathbb{P}}(k,s) &= 0 \ \forall \ k \in \mathbb N \\\\ \zeta_{\mathbb{P}}(2,s) &= \dfrac{\zeta_{\mathbb{P}}(s)^2 + \zeta_{\mathbb{P}}(2s)}{2}. \end{align}\]

**Cite as:**Prime Zeta Function.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/prime-zeta-function/