On Bombieri's asymptotic sieve
The main subject of this paper belongs to the area of analytic number theory. It makes an extensive use of classical techniques and notation that are specific to the subject, together with numerous new tricks. The amount of notation and intricate technicalities contained in the paper explains why the present review only gives a very superficial overview of the results and proofs. The main result of the paper under review consists in an improvement on the error term in Bombieri's asymptotic sieve in the case of a summation over integers having at most two prime factors. More precisely, setting \(\Lambda^{(t)}(n):=\Lambda(n)\log n/(t-1)!\), the author studies the sum \[ \Sigma_\nu(f,X)=\sum_{n\leq X}\Lambda^{(2\nu)}(n)f(n)+\sum_{n\leq X}\Lambda^{(\nu)}\star \Lambda^{(\nu)}(n)f(n)\,, \] where summations are restricted to integers coprime to a fixed integer \(\mathfrak{f}\) and where \(\star\) denotes Dirichlet's convolution. Of particular interest to the author is the case where the sums are performed over a set of integers with at most two prime factors. The weight functions \(f\) that are part of the input may range over several families of nonnegative valued functions (satisfying some extra regularity conditions). The author explains that it is reasonable to hope to get some understanding of \(\Sigma_\nu(f,X)\) only in the case where \(f\) is close to a ``model'' function \(f_0\) (attached to a multiplicative function \(\sigma_0\)). The study of the consequences of an increasing of \(\nu\) plays a fundamental role in the proof. The author notably notices that \(\Lambda^{(\nu)}\star \Lambda^{(\nu)}\) becomes concentrated on integers that are products of two prime powers with similar size as \(\nu\) increases. Moreover, as \(\nu\) increases, the error term obtained by the author improves. A noticeable breakthrough contained in the paper is the following estimate: \[ \sum_{p\leq X}\exp(2i\pi p\alpha)\log p\ll X{\sqrt{q}\over \varphi(q)}\,, \] when \(\alpha = \frac{a}{q}+\beta\), where \(\log q \leq 0.02(\log x)^{1/3}\) and \(|\beta| \leq X^{-1}\exp\big( (\log X)^{1/3} \big)\). As the author points out this estimate is ``best possible'' in the sense that any improvement on the above estimate would imply stronger-than-known results on exceptional zeros of Dirichlet \(L\)-functions.
scientific article; zbMATH DE number 5708636