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.

The value of $\zeta_{\mathbb{P}}(1)$ is the sum of the reciprocals of the primes. This series diverges, but very slowly: $$\sum_{\stackrel{p \in \mathbb{P}}{p < n}} \frac1{p} - \log\big(\log(n)\big) \to M\ \text{ as }\ n\to \infty,$$ 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.

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 $$\begin{aligned} \log\big(\zeta(s)\big) &= - \sum_{p \text{ prime}} \log\left( 1-\frac1{p^s} \right) \\ &= \sum_{p \text{ prime}} \sum_{k=1}^{\infty} \frac{\hspace{1.5mm} \frac{1}{p^{ks}}\hspace{1.5mm} }{k}, \end{aligned}$$ using the Maclaurin series $-\log(1-x) = x+\frac{x^2}2 + \frac{x^3}3 + \cdots.$

Now switching the two sums gives $$\begin{aligned} \log\big(\zeta(s)\big) &= \sum_{k=1}^{\infty} \sum_{p \text{ prime}} \frac{\hspace{1.5mm} \frac{1}{p^{ks}}\hspace{1.5mm} }{k} \\ &= \sum_{k=1}^{\infty} \frac1{k} \sum_{p \text{ prime}} \frac1{p^{ks}} \\ &= \sum_{k=1}^{\infty} \frac{\zeta_{\mathbb{P}}(ks)}{k}. \end{aligned}$$ 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 $x \to 0^+ .$

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.

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 $L_1 = 1, L_2 = 3, L_n = L_{n-1}+L_{n-2}\big).$

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.

Try proving these interesting identities involving generalized prime zeta function:

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