Does the Laplace transform of the Mobius function generating function exist?

[eljose79]

let be the function given by f(x)=w(x)t^-3mu(x) where mu(x) is the Mobius function and w(x)=Sum(1<n<infinite)d(x-n) then my question is...does the Laplace transform of this function exist and is equal to

L[f(x)]=Sum(1<n<Infinite)mu(n)/n^3

 

[eljose79]

sorry i made a mistake it should be L[f(x)]=Sum(1<n<Infinite)mu(n)exp(-sn)/n^3

 

[matt grime]

It isn't a function (from R to R) so asking if its Laplace transform exists as a *function* seems a little moot.

 

[matt grime]

the answer is, mutatis mutandis, yes, by the way, since that infinite sum obviously converges, for some range of s, to a real number. whether or not that is meaningful is a different question

 

[eljose79]

The question is interesting when related the generating function of MOebius function

Sum(n)mu(n)/n^(4-s)=R(4-s) where the sum is from 1 to infinite then according to our formula:

R(4-s)=M[w(x))/x^3] or M^-1[R(4-s)]=w(x)mu(x)/x^3 now integrating from k-1/2 to k+1/2 we have that Int(k-1/2,k+1/2)M^-1[R(4-s)]=mu(k)/k^3