Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparse sets that satisfy the prime number theorem

Olivier Bordellès, Randell Heyman, Dion Nikolic

Abstract

We investigate various sparse sets that satisfy the prime number theorem. The sparsest of these sets, $\{\lfloor x/n^t \rfloor:n \le x\}$, has density approaching $1/x$ as $t$ approaches infinity.

Sparse sets that satisfy the prime number theorem

Abstract

We investigate various sparse sets that satisfy the prime number theorem. The sparsest of these sets, , has density approaching as approaches infinity.
Paper Structure (19 sections, 13 theorems, 69 equations)

This paper contains 19 sections, 13 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Organisation of this Paper
  3. Preparatory Lemmas
  4. Proof of Theorem \ref{['thm:x/n^t']}
  5. Assumption regarding size of $x$
  6. Splitting $\pi$S_⌊xn^t⌋,x$$ into subsums
  7. Basic tools
  8. The sum $S_1(x,t)$
  9. The sum $S_2(x,t)$
  10. The sum $\Sigma_{t,1}(x)$
  11. The sum $\Sigma_{t,2}(x)$
  12. The sum $S_2(x,t)$
  13. The sum $S_3(x,t)$
  14. Conclusion
  15. Proof of Lemma \ref{['lem:o(x)']}
  16. ...and 4 more sections

Key Result

Corollary 1

With the constraints mentioned above, the family of sets $S_{\left\lfloor n^c\right\rfloor,x}$ do not satisfy the prime number theorem. In contrast, the family of sets $S_{\left\lfloor\alpha n + b\right\rfloor,x}$ do satisfy the prime number theorem. Moreover, $D_{\left\lfloor\alpha x+\beta\right\rf

Theorems & Definitions (22)

  • Corollary 1
  • Theorem 1
  • Example 1
  • Conjecture 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 12 more