The special Liouville function displayed as a tridimensional random walk for the integer numbers from 2 to 150001 [La fonction spéciale de Liouville visualisée comme une marche aléatoire tridimensionnelle pour les nombres entiers de 2 à 150001].
The special Liouville function displayed as a tridimensional random walk for the integer numbers from 2 to 150001 [La fonction spéciale de Liouville visualisée comme une marche aléatoire tridimensionnelle pour les nombres entiers de 2 à 150001].
DPD(2) = 1
DPD(3) = 1
DPD(4) = 1
DPD(5) = 1
DPD(6) = 2
DPD(7) = 1
DPD(8) = 1
DPD(9) = 1
(...)
Let's recall that 1 is not a prime number when 2 is the first one (and the only even one...).
DPD(n)
sL(n) = (-1)
Hence:
sL(2) = -1
sL(3) = -1
sL(4) = -1
sL(5) = -1
sL(6) = +1
sL(7) = -1
sL(8) = -1
sL(9) = -1
(...)
X(0) = 0
Y(0) = 0
Z(0) = 0
X(i+1) = X(i) + L(3*i+2)
Y(i+1) = Y(i) + L(3*i+3)
Z(i+1) = Z(i) + L(3*i+4)
(the point {X(0),Y(0),Z(0)} is on top of the picture -white point-, when the colors used {magenta,red,yellow,green,cyan}
are an increasing function of 'i')
with the one of the Liouville function 
.