Some restricted partition functions (Q1310898)

Let \(\| f(q)\|_ A= \sup_{q\in A} | f(q)|\). The problem discussed in this paper is the size of \[ \eta(n,p)= \min_{\textstyle {{{\alpha_ 1, \alpha_ 2,\dots, \alpha_ n\in\mathbb{N}} \atop {p\nmid \alpha_ i}}}} \Biggl\| \prod_{k=1}^ n (1-q^{\alpha_ k}) \Biggr\|_{| q|=1}, \] where \(p\) is prime. Previously, Erdős and Szekeres had raised the problem of estimating the above expression but without the condition \(p\nmid \alpha_ k\). The main result proved here is that \(\eta(n,p)\geq p^{n/(p-1)}\) for all primes \(p\), and that \(\eta(n,p)= O(p^{n/(p-1)})\) holds only for \(p\leq 13\). This distinction between primes \(\leq 13\) and those \(>13\) arises by a study of the product \(\displaystyle{\prod^ n_{{{k=1} \atop {p\nmid k}}}}(1-q^ k)\) which is related to partitions into distinct parts not divisible by \(p\). In the course of his study, the author makes clever use of various trigonometrical sums and products one of which is \(\prod_{k=1}^ n \sin(k\theta+\gamma)\).