[seqfan] Lambert series for Mertens function?
Mats Granvik
mgranvik at abo.fi
Wed Sep 8 15:57:11 CEST 2010
By using x=0.1 as a starting point I imitated the pattern in A176702.
The Lambert series I get for the Mertens function is this:
Sum_{n >= 1} n=1: A002321(n)*x^1/(1-x^1)-x^2/(1-x^2) (+)
n>1:
A002321(n)*(x^n/(1-x^n)-x^(3*2^(n-1))/(1-x^(3*2^(n-1)))-(x^(n+1)/(1-x^(n+1))-x^(n*(n+1))/(1-x^(n*(n+1))))) =
x.
As a Scilab (Matlab clone) program it is:
//Program starts
x=0.1
Mertens = [1;0;-1;-1;-2;-1;-2;-2;-2;-1;-2;-2;-3;-2];
Lambert=zeros(14);
Lambert(1)=Mertens(1)*x^1/(1-x^1)-x^2/(1-x^2);
for n=2:14;
Lambert(n)=Mertens(n)*(x^n/(1-x^n)-x^(3*2^(n-1))/(1-x^(3*2^(n-1)))-(x^(n+1)/(1-x^(n+1))-x^(n*(n+1))/(1-x^(n*(n+1)))));
end
new_x=sum(Lambert)
//Program ends
Is this true?
Best regards,
Mats Granvik
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