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A finite Dirichlet series related to Newman polynomials
Abstract
$$\rm Define\quad F\sb{n}: \IR\sp{n} \to \IR\ by\ F\sb{n}(x\sb1,x\sb2,\cdots,x\sb{n}) = \vert\sum\limits\sbsp{j=1}{n} e\sp{ix\sb{j}}\vert\sp2$$and$$\rm f(n) = {\sup\sb{x\sb1 < x\sb2 < \cdots < x\sb{n}}}\ {\inf\sb{\alpha\in\IR}}\ (F\sb{n}(x\sb1\alpha,x\sb2\alpha,\cdots,x\sb{n}\alpha))\sp{1\over2}.$$To evaluate f(n), it suffices to consider only nonnegative integral values for the $\rm x\sb{j},$ j = $\rm 1,2,\cdots,n,$ with $\rm x\sb1$ = 0. A Newman polynomial is defined to be a complex polynomial P(z) = $\rm\sum\sbsp{j = 1}{n}\ z\sp{x\sb{j}}$ with coefficients all zero or one, and constant term equal to one. Consequently, for integral x$\sb{\rm j}$'s, with x$\sb1$ = 0, we have:$$\rm{\inf\sb{\alpha \in \IR}}\ (F\sb{n}(x\sb1 \alpha,x\sb2 \alpha,\cdots,x\sb{n}\alpha))\sp{1\over2} = {\min\sb{\vert z\vert=1}}\ \vert P(z)\vert.$$ In the process of evaluating f(n), we produce a Newman polynomial with n nonzero terms, with the largest minimum modulus on the unit circle. We specifically calculate f(1), f(2), f(3), and f(4) and determine some computational results for f(5) and f(6).
Subject Area
Mathematics
Recommended Citation
Goddard, Bartley Earl, "A finite Dirichlet series related to Newman polynomials" (1989). ETD collection for University of Nebraska-Lincoln. AAI9019569.
https://digitalcommons.unl.edu/dissertations/AAI9019569