Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Exceptional characters and prime numbers in sparse sets

Jori Merikoski

Abstract

We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove non-trivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form $a^2+b^8$.

Exceptional characters and prime numbers in sparse sets

Abstract

We develop a lower bound sieve for primes under the (unlikely) assumption of infinitely many exceptional characters. Compared with the illusory sieve due to Friedlander and Iwaniec which produces asymptotic formulas, we show that less arithmetic information is required to prove non-trivial lower bounds. As an application of our method, assuming the existence of infinitely many exceptional characters we show that there are infinitely many primes of the form .

Paper Structure

This paper contains 18 sections, 15 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Sketch of the argument
  3. Notations
  4. Acknowledgements
  5. The sieve argument
  6. Preliminaries
  7. Arithmetic information
  8. Initial decomposition
  9. Sum $S_1$
  10. Sum $S_2$
  11. Sum $S_{21}$
  12. Sum $S_{23}$
  13. Sum $S_{22}$
  14. Bounding the error term $S_{222}$
  15. Proof of the upper bound result
  16. ...and 3 more sections

Key Result

Theorem \oldthetheorem

If there are infinitely many exceptional primitive characters $\chi$, then there are infinitely many prime numbers of the form $a^2+b^8$. More precisely, if $L(1,\chi_{D}) \leq \log^{-100} D$, then for $\exp(\log^{10} D) < x <\exp(\log^{16} D)$ we have and

Theorems & Definitions (27)

  • Theorem \oldthetheorem
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 17 more