[seqfan] Where should I submit this?

Mats Granvik matsgranvik at outlook.com
Sun Apr 13 17:21:00 CEST 2014




Two well known arithmetics functions, the von Mangoldt function and
the Möbius function, are found in this matrix "T" whose entries are the Dirichlet
inverse of the Euler totient as follows:
 
T(n,k) = a(GCD(n,k))
 
where "n" is row index and "k" is column index.
 
GCD stands for Greatest Common Divisor, and
"a" is the said Dirichlet inverse of the Euler totient function.
 
starting:
 
a(n) = 1, -1, -2, -1, -4, 2, -6, -1, -2, 4, -10, 2, -12, 6, 8
 
"n" = 1,2,3,4,...
 
T(n,k) is already entered as http://oeis.org/A023900 in the oeis.
 
I had entered this into wikipedia with a link my math stackexchange question
answered by "joriki" and another link to the OEIS page above.
 
However today I found out I had been deleted from Wikipedia by some
user there.
 
The claims of which one I believe is known by the result of Wolfgang Schramm
on GCD matrices and the so called "Fourier-GCD transform" are:
 
Claim 1 (Wolfgang Schramm):
 
Sum_{k=1}^{k=n} T(n,k)*Cos(-2*Pi*k/n)/n = MobiusMu(n)
 
Claim 2 (Mats Granvik, but probably known):
 
Sum_{n=1}^{n=Infinity} = T(n,k)/n = MangoldtLambda(n)
 
which by symmetry of matrix "T" allows for permutation of variables, of course.
 
I think these are important.
 
Also at least a heuristic proof of the prime number theorem is given by showing
that the von Mangoldt function is on average equal to 1.
 
What has not been investigated are the sums in the other direction of the
"Fourier-GCD transform" matrix. They appear to converge to something
and the sequence is very irregular. The first convergent is found here:
 
http://oeis.org/A210617
 
0.4790882572765523426... 
 
 
Where should I submit these two claims? I would like to have it entered into
Wikipedia. Two persons have said that for a proof of Claim 2, the von Mangoldt
function series, Abels theorem is required, but GH from MO at Math Overflow
gave a simpler proof.
 
Best regards,
Mats

 		 	   		  


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