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 value of is the sum of the reciprocals of the primes. This series diverges, but very slowly: where is a constant (called the Meissel-Mertens constant). This is reminiscent of the definition of the Euler-Mascheroni constant .
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 evaluated at ). As is often the case with Euler's arguments, it can be made rigorous with some extra work.
Now switching the two sums gives A generalization of Möbius inversion says that where is the Möbius function, as long as the sums are absolutely convergent (the proof is straightforward). Applying this gives Note that this gives an idea of why diverges at the same speed as , since the term is the only undefined term at and diverges like . In fact, expanding near gives for where is the Euler-Mascheroni constant. The terms go to as
The twin prime conjecture is that there are infinitely many primes such that is also prime. While this is still open, heuristics suggest that it is true and that in fact the function that counts twin primes satisfies where is the twin prime constant
A computation involving Taylor expansions of logarithms (similar to the above one) shows that the constant is related to the values of as follows: The point is that the prime zeta function comes up in evaluating constants involving products over all primes.
Artin's conjecture states that if is an integer that is not a perfect square or , then it is a primitive root modulo for infinitely many As usual, there is a heuristic estimate for the probability that is a primitive root mod a given prime If is squarefree and not congruent to mod , this probability is conjectured to be Artin's constant (For other values of , there is also a conjectural probability that is a rational multiple of Artin's constant.)
A calculation involving taking logs and expanding gives where is the Lucas number defined by
An interesting generalization can be made by summing over inverse of positive integers raised to power which are a product of primes (not necessarily distinct).
The prime zeta function , where is a complex number and is a non-negative integer, is defined by the series where denotes the number of prime factors.
Try proving these interesting identities involving generalized prime zeta function: