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:
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)
If [a Dirichlet] series
, then it also converges for all
. If it diverges for
, then it also diverges for all
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 ?