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A number-theoretic problem concerning pseudo-real Riemann surfaces

Gareth A. Jones, Alexander K. Zvonkin

Abstract

Motivated by their research on automorphism groups of pseudo-real Riemann surfaces, Bujalance, Cirre and Conder have conjectured that there are infinitely many primes $p$ such that $p+2$ has all its prime factors $q\equiv -1$ mod~$(4)$. We use theorems of Landau and Raikov to prove that the number of integers $n\le x$ with only such prime factors $q$ is asymptotic to $cx/\sqrt{\ln x}$ for a specific constant $c=0.4865\ldots$. Heuristic arguments, following Hardy and Littlewood, then yield a conjecture that the number of such primes $p\le x$ is asymptotic to $c'\int_2^x(\ln t)^{-3/2}dt$ for a constant $c'=0.8981\ldots$. The theorem, the conjecture and a similar conjecture applying the Bateman--Horn Conjecture to other pseudo-real Riemann surfaces are supported by evidence from extensive computer searches.

A number-theoretic problem concerning pseudo-real Riemann surfaces

Abstract

Motivated by their research on automorphism groups of pseudo-real Riemann surfaces, Bujalance, Cirre and Conder have conjectured that there are infinitely many primes such that has all its prime factors mod~. We use theorems of Landau and Raikov to prove that the number of integers with only such prime factors is asymptotic to for a specific constant . Heuristic arguments, following Hardy and Littlewood, then yield a conjecture that the number of such primes is asymptotic to for a constant . The theorem, the conjecture and a similar conjecture applying the Bateman--Horn Conjecture to other pseudo-real Riemann surfaces are supported by evidence from extensive computer searches.
Paper Structure (10 sections, 2 theorems, 90 equations, 8 tables)

This paper contains 10 sections, 2 theorems, 90 equations, 8 tables.

Table of Contents

  1. Introduction
  2. Raikov's Theorem
  3. Distribution of $\mathcal{P}$-integers
  4. An alternative evaluation of $c$
  5. The twin primes conjecture
  6. In the footsteps of Hardy and Littlewood
  7. Computations
  8. Some remarks
  9. A similar problem
  10. Appendix: the pseudo-real surfaces

Key Result

Theorem 2.1

Let $F(s)=\sum_{n\ge 1}a_n/n^s$ be a Dirichlet series with non-negative coefficients, converging for ${\rm Re}(s) > 1$. Suppose that $F(s)$ extends analytically at all points on ${\rm Re}(s) = 1$ apart from $s=1$, and that at $s=1$ we can write for some $\alpha\in{\mathbb R}$ and some $H(s)$ holomorphic in the region ${\rm Re}(s)\ge 1$ and nonzero there. Then as $x\to\infty$, with where $\Gamma

Theorems & Definitions (4)

  • Conjecture 1.3
  • Theorem 2.1
  • Theorem 3.1
  • Conjecture 6.1