Golomb's \(\Lambda\)-calculus and the Bateman-Horn conjecture
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 series by the corresponding Dirichlet series. The present authors extend Golomb's method to study the BHC by considering the power series \[ \sum^\infty_{n=1} \Lambda(f_1(n)) \Lambda(f_2(n))\cdots\Lambda(f_k(n)) z^n. \] Here also the proof is valid except for the justification in the general case of the evaluation of a certain limit (although the limit can be verified when \(k= 1\) and \(f_1\) is linear). The paper ends with a discussion of various forms of the Goldbach conjecture subsequently ruled out by the work of E. Landau and of \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Acta Math. 44, 1--70 (1922; JFM 48.0143.04)], and an explanation of how to attempt a proof using Golomb's method. The long list of references includes papers by well known mathematicians from the second half of the nineteenth century.
scientific article; zbMATH DE number 5125710
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