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[] at (6.40000,4.20000) {$u=|z|/(\sqrt2\pi)$};
3 \draw[->] (0.70000,3.20000) -- (12.30000,3.20000);
4 \draw[line width=1pt] (2.63333,3.08000) -- (2.63333,3.32000);
5 \draw[fill=black] (2.63333,3.20000) circle (0.055);
6 \node[] at (2.63333,2.78000) {$1/2$};
7 \node[font=\sffamily\footnotesize] at (2.63333,3.65000) {count 4};
8 \draw[line width=1pt] (6.30000,3.08000) -- (6.30000,3.32000);
9 \draw[fill=black] (6.30000,3.20000) circle (0.055);
10 \node[] at (6.30000,2.78000) {$3/2$};
11 \node[font=\sffamily\footnotesize] at (6.30000,3.65000) {count 8};
12 \draw[line width=1pt] (9.96667,3.08000) -- (9.96667,3.32000);
13 \draw[fill=black] (9.96667,3.20000) circle (0.055);
14 \node[] at (9.96667,2.78000) {$5/2$};
15 \node[font=\sffamily\footnotesize] at (9.96667,3.65000) {count 12};
16 \node[] at (6.40000,2.00000) {$v=|w|/(4\pi^4)$};
17 \draw[->] (0.70000,1.00000) -- (12.30000,1.00000);
18 \draw[line width=1pt] (0.81719,0.88000) -- (0.81719,1.12000);
19 \draw[fill=black] (0.81719,1.00000) circle (0.055);
20 \node[] at (0.81719,0.58000) {$1/16$};
21 \node[font=\sffamily\footnotesize] at (0.81719,1.45000) {count 1};
22 \draw[line width=1pt] (2.19219,0.88000) -- (2.19219,1.12000);
23 \draw[fill=black] (2.19219,1.00000) circle (0.055);
24 \node[] at (2.19219,0.58000) {$81/16$};
25 \node[font=\sffamily\footnotesize] at (2.19219,1.45000) {count 2};
26 \draw[line width=1pt] (11.54219,0.88000) -- (11.54219,1.12000);
27 \draw[fill=black] (11.54219,1.00000) circle (0.055);
28 \node[] at (11.54219,0.58000) {$625/16$};
29 \node[font=\sffamily\footnotesize] at (11.54219,1.45000) {count 3};
30 \node[] at (6.50000,-0.05000) {$n_{D_4}(\sqrt2\pi u)=4\lfloor u+1/2\rfloor,\quad n_{\Delta_4}(4\pi^4 v)=\lfloor v^{1/4}+1/2\rfloor$};
31 \end{tikzpicture}

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