This post contains a brief proof of **Conj. 3** posted on Lee’s Conjecture. To avoid confusion I used as real variable, and as complex.

I proposed a conjecture as follows:

**Conj. 3:**

Let be a set of primes. The supremum of such that diverges is .

Let us define a Prime Zeta Function as follows: .

In fact, the conjecture turned out to be an easy comparison series problem.

Consider the original Riemann Zeta function, that is defined as . It was already proven that is defined on a half plane of such that . The domain was extended over all complex plane except via Analytic Continuation in Riemann’s paper.

Since the conjecture is asking the supremum of , let us only consider the case of real numbers. Obviously, for , as each term in the summand of is also included in , and every term in is positive. Thus, as long as is defined, so is . Hence, the domain of includes a half real line . Therefore, the supremum of where diverges must be ; the conjecture is true.

While studying Introduction to Analytic Number Theory by T. Apostol, I encountered a following theorem. (1976 Edition, pp.233)

**Theorem 11.8:**

If [a Dirichlet] series converges for , then it also converges for all with . If it diverges for , then it also diverges for all with

presented in the theorem is an arithmetical function. Consider a prime detecting function such that for being prime, and for being composite. Then, the Dirichlet series Therefore, we may apply **Theorem 11.8** to deduce the following fact.

We already showed that for any real number converges. Take arbitrarily small , then converges. Thus, by **Theorem 11.8**, for any with also converges. Letting , we can conclude $P(s)$ is defined for all . Thus, the domain of is now extended to the half plane where .

Then, we can come up with a few new problems.

- Can we extend the domain of via Analytic Continuation?
- As we defined Prime Zeta Function, let us define Composite Zeta Function, that is . Will the Composite Zeta Function have the same domain?
- Does Prime Zeta Function have functional equation as Riemann Zeta Function? If so, what are the trivial zeroes and nontrivial zeroes of ?