Weighted sieves and twin prime type equations
By the methods of \textit{J. Chen} [Sci. Sin., Ser. A 16, 157-176 (1973; Zbl 0319.10056)] it is known that the equation \(p+2=P_ 2\) has infinitely many solutions, where p is prime and \(P_ 2\) has at most two prime factors. Chen's methods involve the use of a ``switching principle'' whereby an upper bound is obtained (and used) for the number of solutions of a subsidiary equation \(p+2=p_ 1p_ 2p_ 3\) over appropriate ranges of the primes involved. The authors give a proof of Chen's result, in a quantitatively weaker form, without using the ``switching principle''. Indeed, their methods would deal, with less difficulty, with the analogous equation \[ p_ 1... p_ k+2=P_ 2 \] for which no ``switching'' argument is available. They use a weighted sieve in a form provided by \textit{M. Laborde} [Mathematika 26, 250-257 (1979; Zbl 0429.10028)] and the result of \textit{E. Bombieri}, \textit{J. B. Friedlander} and \textit{H. Iwaniec} [Acta Math. 156, 203-251 (1986; Zbl 0588.10042)] concerning expressions of the form \(\sum_{q\leq x^ c}\lambda (q) E(x,q,a)\) where \(c<4/7\), E is the error term in the Prime Number Theorem for arithmetic progressions and \(\lambda\) (q) is a ``well-factorable'' coefficient. Since the weighted sieve leads to coefficients which are not exactly ``well-factorable'' an extension of this result has to be developed.
scientific article; zbMATH DE number 4118460
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