Mertens function identity. Scilab, Matlab

// Just for fun, this program manipulates (A_eps)*n^(1/2+eps) into the Mertens function. This proves nothing though, see the text at the end.

x=7; // size of matrix
T=zeros(x,x);

A_eps = 1; // can be any value
eps = 0.001; // can be any value

// Values for second column
for n=1:x;
T(n,2)=(A_eps)*((n-1)^(1/2+eps)); // Expression from Doron Zeilbergers homepage.
end

// Value for first element
T(1,1)=1;

// Values for first column
for n=2:x;
T(n,1)=T(n-1,2)+T(n,2);
end

// Values for the table from the third column onwards
for n=3:x;
for k=3:n;
s = 0;
for i=1:k-1;
//recursive definition of divisibility without using the mod function
s=s+T(n-i,k-1)-T(n-i,k);
end
T(n,k)=s;
end
end

// Matrix A which first column beginning at A(2,1) is the second column in matrix T.
A=zeros(x,x);
for n=1:x;
for k=1:x-1;
A(n,k)=T(n,k)-T(n,k+1);
end
end
for n=1:x;
A(n,x)=T(n,x);
end
A

// Matrix B is the inverse of matrix A. First column of matrix B is the Mertens function for almost any values in second column of matrix T. This procedure could perhaps be called an identity.
B=inv(A)
// Mats Granvik, email: mats.granvik(at)abo.fi