On sums of coefficients of polynomials related to the Borwein conjectures (Q2070395)

In 1990, Peter Borwein (see \textit{G. E. Andrews} [J. Symb. Comput. 20, 487--501 (1995; Zbl 0849.68062)]) conjectured for the coefficients of certain polynomials \(\sum_{j\ge 0}a_jq^j\) that \(a_j\ge 0\) if \(j \equiv 0 \mod p\), and \(a_j\le 0\) otherwise, where \(p=3\) or \(p=5\). In the paper under review, the authors consider the more general polynomial \[ T_{p,s,n}(q)=\prod _{j=0}^n \prod _{k=1}^{p-1}\left( 1 - q^{pj+k}\right)^s \] for positive integers \(s, n\), and odd prime \(p\). The special cases \(s=1,\; p=3;\; s=2,\; p=3;\; s=1,\; p=5\) give the polynomials in the First, Second, and Third Borwein conjecture, respectively. The authors prove asymptotic formulae for several partial sums of the coefficients of \(T_{p,s,n}(q)\). In the special cases \(p=3, 5\) the signs of these sums are consistent with the Borwein conjectures. The proofs extend the method of \textit{J. Li} [Int. J. Number Theory 16, 1053--1066 (2020; Zbl 1469.11410)] which was based on a sieving principle of \textit{J. Li} and \textit{D. Wan} [Sci. China Math. 53, 2351--2362 (2010; Zbl 1210.05010)]. Similar sums have been studied earlier by \textit{A. Zaharescu } [Ramanujan J. 11, 95--102 (2006; Zbl 1161.11313)]. The authors also improve on the error terms in the asymptotic formulae due to Li and Zaharescu. An analytic proof of the First Borwein conjecture was published by \textit{C. Wang} [``An analytic proof of the Borwein conjecture'', Preprint, \url{arXiv:1901.10886}; Adv. Math. 394, Article ID 108028, 54 p. (2022; Zbl 1479.05032)]. Using a result of \textit{P. Borwein} [J. Number Theory 45, 228--240 (1993; Zbl 0788.11043)] the authors also obtain a logarithmic asymptotic estimate (resp. a lower bound) for the maximum of the absolute values of the coefficients in \(T_{p,s,n}(q)\) as \(n\to \infty\) for \(p=2,3,5,7,11,13\) (resp. for \(p>15\)).