G(1,1) p=4          = np.float64(0.11062653532614816)
zeta(3)/zeta(4) - 1 = np.float64(0.11062653532614797)
area pi*G(1,1)      = np.float64(0.34754351067271877)  triangle area 2-pi/2 = 0.42920367320510344  ratio 0.8097402989993476
sum_{q=2}^{2e5} phi(q)/q^4 = 0.11062653531854906
brute vs lattice 1 1 0.11062456248889171 0.11062653532614816
brute vs lattice 1 0.3 0.00669076437781504 0.006691114951221562
brute vs lattice 0.7 2.0 0.15935228093539308 0.15935894236901949
recursion p=2.5 a=1 b=1 rel.err=3.52e-16
recursion p=4.0 a=1 b=1 rel.err=5.02e-16
recursion p=7.0 a=1 b=1 rel.err=0.00e+00
recursion p=2.5 a=0.2 b=3 rel.err=1.63e-16
recursion p=4.0 a=0.2 b=3 rel.err=2.14e-16
recursion p=7.0 a=0.2 b=3 rel.err=6.42e-16
recursion p=2.5 a=5 b=0.01 rel.err=5.35e-16
recursion p=4.0 a=5 b=0.01 rel.err=1.15e-16
recursion p=7.0 a=5 b=0.01 rel.err=7.25e-16
Lemma 4 check: 5550 second differences on 150 random arcs; most negative (relative) = 0.000e+00
gap(1.00,1.00) p=4.0 k=2 starts=12  max (Phi-G)/G = -5.018e-16
gap(1.00,1.00) p=4.0 k=3 starts=12  max (Phi-G)/G = -2.509e-16
gap(1.00,1.00) p=4.0 k=4 starts=12  max (Phi-G)/G = 7.527e-16
gap(1.00,1.00) p=4.0 k=5 starts=12  max (Phi-G)/G = -5.018e-16
gap(1.00,1.00) p=2.5 k=2 starts=12  max (Phi-G)/G = -2.344e-16
gap(1.00,1.00) p=2.5 k=3 starts=12  max (Phi-G)/G = -3.516e-16
gap(1.00,1.00) p=2.5 k=4 starts=12  max (Phi-G)/G = 0.000e+00
gap(1.00,1.00) p=2.5 k=5 starts=12  max (Phi-G)/G = 0.000e+00
gap(1.00,1.00) p=6.0 k=2 starts=12  max (Phi-G)/G = -3.604e-16
gap(1.00,1.00) p=6.0 k=3 starts=12  max (Phi-G)/G = 5.226e-15
gap(1.00,1.00) p=6.0 k=4 starts=12  max (Phi-G)/G = 0.000e+00
gap(1.00,1.00) p=6.0 k=5 starts=12  max (Phi-G)/G = 0.000e+00
gap(1.00,0.25) p=4.0 k=2 starts=12  max (Phi-G)/G = 2.098e-16
gap(1.00,0.25) p=4.0 k=3 starts=12  max (Phi-G)/G = 7.133e-15
gap(1.00,0.25) p=4.0 k=4 starts=12  max (Phi-G)/G = 1.594e-14
gap(1.00,0.25) p=4.0 k=5 starts=12  max (Phi-G)/G = 0.000e+00
gap(1.00,0.25) p=2.5 k=2 starts=12  max (Phi-G)/G = 5.251e-15
gap(1.00,0.25) p=2.5 k=3 starts=12  max (Phi-G)/G = 9.377e-16
gap(1.00,0.25) p=2.5 k=4 starts=12  max (Phi-G)/G = 2.813e-15
gap(1.00,0.25) p=2.5 k=5 starts=12  max (Phi-G)/G = 1.519e-14
gap(1.00,0.25) p=6.0 k=2 starts=12  max (Phi-G)/G = 5.770e-14
gap(1.00,0.25) p=6.0 k=3 starts=12  max (Phi-G)/G = 2.308e-15
gap(1.00,0.25) p=6.0 k=4 starts=12  max (Phi-G)/G = 3.462e-15
gap(1.00,0.25) p=6.0 k=5 starts=12  max (Phi-G)/G = 1.449e-14
gap(0.30,2.00) p=4.0 k=2 starts=12  max (Phi-G)/G = 3.783e-14
gap(0.30,2.00) p=4.0 k=3 starts=12  max (Phi-G)/G = 4.250e-15
gap(0.30,2.00) p=4.0 k=4 starts=12  max (Phi-G)/G = 4.038e-15
gap(0.30,2.00) p=4.0 k=5 starts=12  max (Phi-G)/G = 2.338e-15
gap(0.30,2.00) p=2.5 k=2 starts=12  max (Phi-G)/G = 4.129e-16
gap(0.30,2.00) p=2.5 k=3 starts=12  max (Phi-G)/G = 4.129e-16
gap(0.30,2.00) p=2.5 k=4 starts=12  max (Phi-G)/G = 2.753e-16
gap(0.30,2.00) p=2.5 k=5 starts=12  max (Phi-G)/G = 5.093e-15
gap(0.30,2.00) p=6.0 k=2 starts=12  max (Phi-G)/G = 1.596e-16
gap(0.30,2.00) p=6.0 k=3 starts=12  max (Phi-G)/G = 1.596e-16
gap(0.30,2.00) p=6.0 k=4 starts=12  max (Phi-G)/G = 4.485e-14
gap(0.30,2.00) p=6.0 k=5 starts=12  max (Phi-G)/G = 9.576e-16
