Merit factors of polynomials formed by Jacobi symbols (Q2715661)

This is another contribution to the literature regarding the maximum modulus of polynomials on the unit circle. The authors consider polynomials of the form NEWLINE\[NEWLINEf_t(z)= \sum^{N-1}_{n=0} \left({n+t \over N}\right) z^nNEWLINE\]NEWLINE whose coefficients are Jacobi symbols with values either \(0\) or \(\pm 1\), and \(1\leq t\leq N\). They obtain exact formulas for the fourth power of the \(L_4\) norm of these polynomials in two special cases, when \(N\) is a product of twin primes, and when \(N=pq\), with \(p,q\) odd primes such that \(p=q+4\) and \(p\equiv 3 \pmod 4\). This extends earlier work of \textit{T. Høholdt}, \textit{J. M. Jense} and \textit{H. E. Jensen} [IEEE Trans. Inf. Theory 37, 617-626 (1991; Zbl 0731.94011)]. NEWLINENEWLINENEWLINEMore generally, when \(N\) is the product of \(r\) distinct primes they obtain an asymptotic formula for the fourth power of the \(L_4\) norm, NEWLINE\[NEWLINE\|f_t\|^4_4= \textstyle {5\over 3}N^2-4Nt+8t^2+O (N^{2+ \varepsilon} p_1^{-1}),NEWLINE\]NEWLINE where \(p_1\) is the smallest of the \(r\) primes.