On the computation of some singular series (Q2857870)

Let \(\theta\) be an irrational algebraic number and let \(I\) be an interval in \([0, 1)\). Let \(P = P(\theta; I)\) denote the set of primes \(p\) such that \(\theta p\) lies in \(I\) modulo one. This paper is a part of a series of several papers, in which the authors study the Waring-Goldbach problem with primes from \(P\). Let \(R_{s,k}(n)\) denote the number of representations of an integer \(n\) as the sum of \(s\) \(k\)th powers of primes, and let \(R_{s,k}^*(n)\) be the number of such representations when the primes are restricted to the set \(P\). To prove the existence of restricted representations, one typically uses Fourier analysis to show that NEWLINE\[NEWLINE R_{s,k}^*(n) \sim \kappa_s(n; \theta, I) R_{s,k}(n) \qquad \text{as } n \to \infty, NEWLINE\]NEWLINE where \(\kappa_s(n; \theta, I) \gg 1\); one then applies known estimates for the classical problem. The coefficient \(\kappa_s\) in the above formula takes the form of an infinite series, namely NEWLINE\[NEWLINE \kappa_s(n; \theta, I) = \sum_{m \in \mathbb Z} \frac{\sin^s(\pi|I|m)}{\pi^sm^s} \exp(2\pi im(\theta n - s\mu)), NEWLINE\]NEWLINE where \(|I|\) and \(\mu\) are, respectively, the length and the midpoint of the interval \(I\). In the paper under review, the authors study the series \(\kappa_s(n; \theta, I)\) and establish finite expansions in terms of the fractional parts of \(\theta n + j|I|\), \(j=0,1,\dots,s\). They also present a couple of examples, in which they use their formulas to analyze the series \(\kappa_s(n; \theta, I)\) when \(s = 2,3\) -- the cases relevant to the study of the Goldbach problem with primes from \(P\).