A note on the Waring-Goldbach problem
For a positive integer \(k\) and a prime \(p\), let \(\nu_p(k)\) denote the \(p\)-adic valuation of \(k\), i.e., \(\nu_p(k)\) is the largest integer \(\nu\) such that \(p^{\nu} \mid k\). Let \(\gamma(k, p) = \nu_p(k)+1\) if either \(p\) or \(k\) is odd, and let \(\gamma(k, 2) = \nu_2(k)+2\) if \(k\) is even. For a positive integer \(k\), we define \(K = K(k) = \prod_{(p-1) \mid k} p^{\gamma(k,p)}\), the product being over all the primes \(p\) such that \((p-1) \mid k\). In simplest terms, the Waring-Goldbach problem for \(k\)th powers is concerned with estimating the least integer \(s\) such that all sufficiently large integers \(n\) can be represented in the form \[ p_1^k + p_2^k + \dots + p_s^k = n, \eqno{(1)} \] where \(p_1, \dots, p_s\) are prime numbers. However, due to \(p\)-adic solubility obstructions, this simple formulation requires some fine-tuning. To avoid degeneracy, one usually restricts \(n\) to certain congruence classes that are ``admissible'' for \(s\)-fold sums of \(k\)th powers of primes. For example, when dealing with sums of two primes, one restricts \(n\) to the even integers; when dealing with sums of three squares of primes, one restricts \(n\) to the set of integers \(n \equiv 3 \pmod {24}\) and such that \(5 \nmid n\). Results for general \(k\) are usually stated in terms of the function \(H(k)\) defined as the least \(s\) such that (1) is soluble in primes \(p_1, \dots, p_s\) for all sufficiently large \(n \equiv s \pmod K\), where \(K = K(k)\) is the integer defined in the last paragraph. For example, the currently best bound for large powers, announced recently by T. D. Wooley, states that \(H(k) \leq k(4\log k + 2)\) when \(k \geq 10\). In the paper under review, the author is concerned with the solubility of (1) without imposing local conditions on \(n\). To this end, he introduces two analogues of the function \(H(k)\) defined above: \(H_{\text{all}}(k)\) denotes the least \(s\) such that (1) is soluble in primes \(p_1, \dots, p_s\) for all sufficiently large \(n\); \(H_{\text{mod}}(k)\) denotes the least \(s\) such that (1) is soluble over \(\mathbb Z_K\) in primes \(p_1, \dots, p_s\) for all \(n \in \mathbb Z_K\). The main result of the paper is the bound \[ H_{\text{mod}}(k) \leq \sum_{(p-1) \mid k} (p-1) + \frac {14}3 k \leq k \big( e^{\gamma}\log\log k + 14/3 + O\big( (\log k)^{-1/2} \big) \big), \eqno{(2)} \] where \(\gamma\) is Euler's constant. When combined with Wooley's bound on \(H(k)\), this yields the general bound \[ H_{\text{all}}(k) \leq k \big( 4\log k + e^{\gamma}\log\log k + 20/3 + O\big( (\log k)^{-1/2} \big) \big) \] in the ``unrestricted'' Waring-Goldbach problem. Further, the author shows that \[ H_{\text{all}}(k) \geq H_{\text{mod}}(k) + k + 1 \geq \sum_{(p-1) \mid k} (p-1) + k + 1, \] and he uses this inequality to provide some heuristics in support of the conjecture that \(H_{\text{all}}(k)\) grows faster than linearly.
scientific article; zbMATH DE number 5634890