Products of shifted primes. Multiplicative analogues of Goldbach's problems. II (Q1265266)

In Part I of this paper [\textit{P. D. T. A. Elliott}, J. Reine Angew. Math. 463, 169-216 (1995; Zbl 0817.11042)], the author formulated some conjectures concerning the representability of rational numbers \( {r \over s} \) (when \( r \leq s \) are ``small'' compared with \( N \)) in the form of a product over shifted primes: \[ \tfrac {r}{s}= \prod_i (N-p_i)^{\varepsilon_i}, \qquad \varepsilon_i \in \{-1, 1\}. \] In the paper under review the author contributes to the third (and weakest conjecture) and proves: There exists an integer \( k \) so that, for a given \( c > 0 \) and for \( N > N_0(c), \) every integer \( m\), \(1 \leq m \leq (\log N)^c\), \(\gcd(m, N) = 1\), has a product representation \[ m^k = \prod_{\substack{ p \leq {1 \over 2} N,\\ p \text{ prime}}} (N-p)^{d_p}, \] where \( d_p \) are integers, and there exists an integer \( k \) and a constant \( \gamma > 0 \) so that every prime \(p\), \(2 \leq p \leq N^\gamma\), \(\gcd (p, N) = 1 \), with at most one exception allows a product representation \[ p^k = \prod_{\substack{ q\leq {1\over 2} N,\\ q \text{ prime}}} (N-q)^{d_q}. \] The proofs need heavy machinery from analytic number theory (harmonic analysis, Bombieri-Vinogradov theorem, Brun-Titchmarsh theorem, sieve estimates).