List of Lee’s Conjectures

If a solution of the unsolved conjectures already exists, please leave a comment.

 

Conj. 1: (Solved)
For any non-negative real number r > 0, there exists a set, K of which strict Hausdorff Dimension is equal to r.

Fields: Measure Theory, Real Analysis
Proposed: 02.24.15
Solved: 03.16.15
The general outline is presented in the post, CONSTRUCTING A SET WITH ARBITRARY STRICT HAUSDORFF DIMENSION

 

Conj. 2: (Unsolved)
Let p be a prime. Let r_p denote the smallest primitive root of (\mathbb{Z}/p\mathbb{Z})^\times. Then, \displaystyle \liminf_{p \to \infty} r_p is bounded, while \displaystyle \limsup_{p \to \infty} r_p diverges.

Fields: Elementary Number Theory
Proposed: 10.28.14

 

Conj. 3: (Solved)
Let \mathbb{P} be a set of primes. The supremum of s such that \displaystyle \sum_{p \in \mathbb{P}} \frac{1}{p^x} diverges is x = 1.

Fields: Analytic Number Theory
Proposed: 02.26.16
Solved: 03.17.16
The general outline is presented in the post, The Half-Plane of Convergence of Zeta Function, and it begot a new conjecture, Conj. 4.

 

Conj. 4: (Unsolved)
Let us define Prime Zeta Function as \displaystyle P(s) = \sum_{p \in \mathbb{P}} \frac{1}{p^s} where \mathbb{P} is a set of primes. The domain of P(s) is all complex plane except s = 1 like Riemann Zeta Function.

Fields: Analytic Number Theory
Proposed: 03.17.16

 

 

The post is still in progress.

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List of Lee’s Conjectures