Paper IV: Cyclic Analytic Functions, Reciprocal Families, and Spectral Zero Data

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1 \begin{tikzpicture}[x=1cm,y=1cm,>=Latex,font=\sffamily\small,line cap=round,line join=round]
2 \node[anchor=west] at (0.35000,4.30000) {denominator};
3 \node[] at (8.00000,4.30000) {$1-w/24+w^2/40320+\cdots$};
4 \draw[black!15] (0.35000,3.87000) -- (12.30000,3.87000);
5 \node[anchor=west] at (0.35000,3.35000) {negative logarithm};
6 \node[] at (8.00000,3.35000) {$w/24+(17/20160)w^2+\cdots$};
7 \draw[black!15] (0.35000,2.92000) -- (12.30000,2.92000);
8 \node[anchor=west] at (0.35000,2.40000) {zero moments};
9 \node[] at (8.00000,2.40000) {$\sigma_1=1/24,\quad\sigma_2=17/10080$};
10 \draw[black!15] (0.35000,1.97000) -- (12.30000,1.97000);
11 \node[anchor=west] at (0.35000,1.45000) {reciprocal};
12 \node[] at (8.00000,1.45000) {$1+w/24+69w^2/40320+\cdots$};
13 \draw[black!15] (0.35000,1.02000) -- (12.30000,1.02000);
14 \node[anchor=west] at (0.35000,0.50000) {tangent};
15 \node[] at (8.00000,0.50000) {$z^3/3!+34z^7/7!+\cdots$};
16 \node[] at (6.50000,-0.55000) {$69=8!\,\frac{\sigma_1^2+\sigma_2}{2},\qquad34=4\cdot7!\,\sigma_2$};
17 \end{tikzpicture}

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