Representation of natural numbers as the sum of values of polynomials of primes (Q1916664)

Let \(f(x)\) be an integer-valued polynomial of degree \(n\) with positive leading coefficient and having no fixed prime divisor. Denote \(E_f(p) =f(p_1)+ \cdots+ f(p_s)\). \textit{L.-K. Hua} [Additive Primzahltheorie, Teubner (1959; Zbl 0088.25503)] established an asymptotic formula for the number of solutions of \(E_f(p) =N\), if \(s\) is large enough (e.g. if \(s \geq 2^n+1\) when \(2\leq n\leq 10)\). The formula involves a ``singular series'' \(\sigma_s(N)\). Let \(l(f)\) be the smallest natural number \(s\) such that \(\sigma_s (N)>0\) for all \(N\) satisfying the congruence conditions necessary mod powers of primes \(\leq n+1\), and let \(\beta_n= \max_f l(f)\), taken over \(f\) of degree \(n\). The article determines \(\beta_n\), with an ambiguity of at most 1. In particular \[ \beta_n\leq 3^{\alpha_n} +2,\quad \text{where} \quad \alpha_n= \bigl[ (n+ 1)/2 \bigr]+ \nu_3\bigl(n! /[n/2]! \bigr), \] in a standard notation. The treatment of \(\beta_n\) involves a corresponding number \(H^*_n\) defined in terms of the congruence \(E_f(p) \equiv N \pmod {p^k}\). Results on \(H^*_n\) are taken from two previous articles of the author [Dep. VINITI No. A-667-B91, Moscow (1991)], [Dep. VINITI No. 4502-B91, Moscow (1991)], both in Russian.