Golomb's \(\Lambda\)-calculus and the Bateman-Horn conjecture
type: Item
instance of: scholarly article
title: Golomb's \(\Lambda\)-calculus and the Bateman-Horn conjecture
zbMATH Open document ID: 1180.11032
author: Marc Hindry
author: T. Rivoal
published in: L'Enseignement Mathématique. 2e Série
publication date: 2007-02-13T00:00:00Z
review text: For \(1\leq i\leq k\), let \(f_i\) be a polynomial in \(\mathbb Z[X]\) with degree \(h_i\geq 1\), positive leading coefficient and irreducible over \(\mathbb Q\), and suppose \(f_i\# f_j\) if \(i\neq j\). Put \(f= f_1 f_2\cdots f_k\), \(\underline f=(f_1,\dots, f_k)\), and assume that for each prime \(p\), there exists \(n\) such that \(p\nmid f(n)\). Let \(\pi_{\underline f}(x)\) denote the number of integers \(n\) with \(1\leq n\leq x\) such that \(f_1(n),f_2(n),\dots, f_k(n)\) are simultaneously primes. A well known conjecture (Hypothesis H) due to A. Schinzel [see \textit{A. Schinzel} and \textit{W. Sierpiński}, Acta Arith. 4, 185--208 (1958; Zbl 0082.25802), erratum 5, 259 (1958)] states that \(\pi_{\underline f}(x)\to\infty\) as \(x\to\infty\). A more precise conjecture (the BHC), based on a heuristic argument, was subsequently made by \textit{P. T. Bateman} and \textit{R. A. Horn} [Math. Comput. 16, 363--367 (1962; Zbl 0105.03302)] who predicted that as \(x\to\infty\) \[ \pi_{\underline f}(x)\sim {C(\underline f)\over h_1h_2\cdots h_k} {x\over\log^kx},\quad C(\underline f)= \prod_p\Biggl(\Biggl(1- {1\over p}\Biggr)^{-k}\Biggl(1- {N_f(p)\over p}\Biggr)\Biggr) \] with \(N_f(p)=\#\{1\leq n\leq p: f(n)\equiv 0\pmod p\}< p\). This includes the Hardy-Littlewood conjecture on primes of the form \(n^2+ 1\). The twin prime conjecture is also a special case of this obtained when \(\underline f= n(n+2)\), and, in an attempt to establish it, \textit{S. W. Golomb} [J. Number Theory 2, 193--198 (1970; Zbl 0198.37601)] developed a method involving the power series \(\sum^\infty_{n= 1}\Lambda(2n-1)\Lambda(2n-1) z^{2n}\) in a neighbourhood of \(z= 1\), where \(\Lambda\) denotes the von Mangoldt function. The proof only failed at one point where the evaluation of a certain limit could not be justified. Recently [Kluwer Int. Ser. Eng. Comput. Sci. 726, 133--154 (2003; Zbl 1054.11050)] \textit{K. Conrad} adapted Golomb's method for this type of problem by replacing the power ser…
Mathematics Subject Classification ID: 11N32
Mathematics Subject Classification ID: 11M45
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 5125710
zbMATH Keywords: Schinzel's Hypothesis H
zbMATH Keywords: Bateman-Horn conjecture
zbMATH Keywords: Golomb's \(\Lambda\)-calculus
reviewed by: Eira J. Scourfield
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5125710