Something which has not been varied so far is the \(s\) parameter in \(\theta(s)_m\). The Reimann zeta function has interesting properties for \(s= \frac{1}{2} + ix\). For now, let \(x=0\) such that \(s\) is a purely real argument. Then for \(s=\frac{1}{2}\) and \(s=\frac{1}{3}\) we generate the series
\[\begin{array}{| c | c | c | | c | c | | c | c |} \hline m & \theta(2)_m & \Delta\theta(2)_m & & \theta(\frac{1}{2})_m & \Delta\theta(\frac{1}{2})_m & & \theta(\frac{1}{3})_m & \Delta\theta(\frac{1}{3})_m \\ \hline 1 & 1 & 1 & & 1 & 1 & & 1 & 1\\ 2 & 2 & 1 & & 3 & 2 & & 3 & 2\\ 3 & 3 & 1 & & 6 & 3 & & 7 & 4\\ 4 & 5 & 2 & & 10 & 4 & & 11 & 4\\ 5 & 6 & 1 & & 14 & 4 & & 16 & 5\\ 6 & 7 & 1 & & 20 & 6 & & 23 & 7\\ 7 & 8 & 1 & & 25 & 5 & & 30 & 7\\ 8 & 10 & 2 & & 32 & 7 & & 40 & 10\\ 9 & 12 & 2 & & 40 & 8 & & 48 & 8\\ 10 & 13 & 1 & & 47 & 7 & & 57 & 9\\ 11 & 14 & 1 & & 53 & 6 & & 68 & 11\\ 12 & 16 & 2 & & 62 & 9 & & 81 & 13\\ 13 & 17 & 1 & & 70 & 8 & & 93 & 12\\ 14 & 18 & 1 & & 80 & 10 & & 105 & 12\\ 15 & 19 & 1 & & 90 & 10 & & 118 & 13\\ 16 & 22 & 3 & & 102 & 12 & & 133 & 15\\ 17 & 23 & 1 & & 110 & 8 & & 146 & 13\\ 18 & 25 & 2 & & 121 & 11 & & 164 & 18\\ 19 & 26 & 1 & & 131 & 10 & & 180 & 16\\ 20 & 28 & 2 & & 144 & 13 & & 195 & 15\\ \hline \end{array}\]