The special Liouville function displayed as a bidimensional random walk for the integer numbers from 2 to 100001 [La fonction spéciale de Liouville visualisée comme une marche aléatoire bidimensionnelle pour les nombres entiers de 2 à 100001].




This picture displays the special Liouville function as a bidimensional random walk. For each integer number 'n' the function DPD(n) gives the number of different prime divisors; for example:
                    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...).

The special Liouville function sL(n) equals:
                                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
                    (...)


Then one defines the following dynamics:
                    X(0) = 0
                    Y(0) = 0
                    X(i+1) = X(i) + L(2*i+2)
                    Y(i+1) = Y(i) + L(2*i+3)
(the point {X(0),Y(0)} is on the right of the picture -white point-, when the colors used {magenta,red,yellow,green,cyan} are an increasing function of 'i')

It is useful to compare this trajectory with the one of the Liouville function .

See the superposition of three different computations: the integer numbers from 2 to 200001 (Red), 2 to 400001 (Green) and 2 to 800001 (Blue) and another superposition of three different computations: the integer numbers from 2 to 400001 (Red), 2 to 800001 (Green) and 2 to 1600001 (Blue) .


(CMAP28 WWW site: this page was created on 01/10/2010 and last updated on 03/03/2016 07:42:30 -CET-)



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