On sums of coefficients of polynomials related to the Borwein conjectures

Peter Borewein empirically discovered quite a number of mysteries involving sign patterns of coefficients of polynomials of the form $f_{p,s,n}(q):=\prod_{j=0}^{n} \prod_{k=1}^{p-1} (1-q^{pj+k})^{s}$ ($p$ a prime and $s,n \in \mathbb{N}$). In the case $(p,s) \in \{(3,1), (3,2)\}$, he conjectured that the coefficients follow a repeating + - - pattern, and in the case $(p,s)=(5,1)$, it was conjectured that the coefficients follow a repeating + - - - - sign pattern. We consider a weaker problem of finding the signs of partial sums of coefficients along some arithmetic progressions. We use a combinatorial sieving principle by Li-Wan and elementary character theory to asymptotically estimate and find the signs of these partial sums. We find that the signs of these partial sums are compatible with the sign pattern in Borewein's conjectures. This is based on joint work with Ankush Goswami.