On the asymptotic formula for the number of representations of numbers as the sum of a prime and a \(k\)-th power (Q1318934)

The author studies the generalization of a well-known conjecture of \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Acta Math. 44, 1-70 (1923; JFM 48.0143.04)] on the asymptotic formula for the number of representations of a natural number \(n\) as a sum of a prime and a \(k\)-th power, provided that the polynomial \(x^ k-n\) is irreducible over \(\mathbb{Z}\). It is shown that the conjectured asymptotic formula holds for ``almost all'' integers. The most difficult point of the proof is the treatment of the singular series and the author follows the work of \textit{R. J. Miech} [Trans. Am. Math. Soc. 130, 494-512 (1968; Zbl 0164.352)] who studied the case \(k=2\). In order to show that the singular series can be approximated with the corresponding infinite product (with an acceptable number of exceptions), one needs some information on the behaviour of the function \(\zeta(s)/ \zeta_ n(s)\) near the line \(\sigma=1\), where \(\zeta_ n\) denotes the Dedekind \(\zeta\)-function of the field \(\mathbb{Q} (n^{1/k})\). In the case \(k=2\), \(\zeta/ \zeta_ n\) is the Dirichlet \(L\)-function associated to a primitive real character, unless \(n\) is a perfect square, and this enabled Miech to use Bombieri's zero density estimate. The author proves an appropriate density theorem for the zeros of the functions \(\zeta/ \zeta_ n\). The details of the proof are going to appear elsewhere.