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 \draw[black!13] (0.80000,0.60000) -- (0.80000,4.20000);
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7 \node[font=\sffamily\footnotesize] at (4.05714,0.29000) {$3$};
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9 \node[font=\sffamily\footnotesize] at (5.68571,0.29000) {$4$};
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11 \node[font=\sffamily\footnotesize] at (7.31429,0.29000) {$5$};
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13 \node[font=\sffamily\footnotesize] at (8.94286,0.29000) {$6$};
14 \draw[black!13] (10.57143,0.60000) -- (10.57143,4.20000);
15 \node[font=\sffamily\footnotesize] at (10.57143,0.29000) {$7$};
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17 \node[font=\sffamily\footnotesize] at (12.20000,0.29000) {$8$};
18 \draw[black!13] (0.80000,1.00000) -- (12.20000,1.00000);
19 \node[anchor=east,font=\sffamily\footnotesize] at (0.64000,1.00000) {$-8$};
20 \draw[black!13] (0.80000,1.80000) -- (12.20000,1.80000);
21 \node[anchor=east,font=\sffamily\footnotesize] at (0.64000,1.80000) {$-6$};
22 \draw[black!13] (0.80000,2.60000) -- (12.20000,2.60000);
23 \node[anchor=east,font=\sffamily\footnotesize] at (0.64000,2.60000) {$-4$};
24 \draw[black!13] (0.80000,3.40000) -- (12.20000,3.40000);
25 \node[anchor=east,font=\sffamily\footnotesize] at (0.64000,3.40000) {$-2$};
26 \draw[black!13] (0.80000,4.20000) -- (12.20000,4.20000);
27 \node[anchor=east,font=\sffamily\footnotesize] at (0.64000,4.20000) {$0$};
28 \draw[->] (0.80000,0.60000) -- (12.35000,0.60000);
29 \draw[->] (0.80000,0.60000) -- (0.80000,4.35000);
30 \draw[line width=1pt] plot coordinates {(0.80000,3.59685) (2.42857,3.23383) (4.05714,2.85954) (5.68571,2.48068) (7.31429,2.10005) (8.94286,1.71875) (10.57143,1.33719) (12.20000,0.95555)};
31 \draw[fill=black] (0.80000,3.59685) circle (0.055);
32 \draw[fill=black] (2.42857,3.23383) circle (0.055);
33 \draw[fill=black] (4.05714,2.85954) circle (0.055);
34 \draw[fill=black] (5.68571,2.48068) circle (0.055);
35 \draw[fill=black] (7.31429,2.10005) circle (0.055);
36 \draw[fill=black] (8.94286,1.71875) circle (0.055);
37 \draw[fill=black] (10.57143,1.33719) circle (0.055);
38 \draw[fill=black] (12.20000,0.95555) circle (0.055);
39 \node[] at (6.50000,4.90000) {$\log_{10}(1-R_n),\quad R_n=\frac{P_{2,n}(1)}{2(2n)!(2/\pi)^{2n+1}}$};
40 \node[] at (6.50000,-0.25000) {Each ratio is enclosed with a rational Machin-formula bound for $\pi$.};
41 \end{tikzpicture}

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