02/27/2023, 01:18 PM

Since we are talking about special dirichlet series , I want to mention this special function again

f(s) = sum (a_n)^(-s)

where a_n is the binary partition function.

(binary partition zeta function or so )

since the binary partition grows faster than polynomial ( yet much slower than exp or even semi-exp ,more like n^ln(n) )

we get that

f(s) converges for Re(s) > 0.

Typical questions occur ; zero's , continuation to Re(s) < 0 , reflection ? natural boundary ?? etc

Im not sure if we have a summability method yet for Q = a_1 + a_2 + a_3 + ...

Can we just go around the pole at 0 , or are there issues ??

Does it have number theoretical or combinatorical meanings ?

regards

tommy1729

" Truth is what does not go away when you stop believing in it "

Tom Marcel Raes

f(s) = sum (a_n)^(-s)

where a_n is the binary partition function.

(binary partition zeta function or so )

since the binary partition grows faster than polynomial ( yet much slower than exp or even semi-exp ,more like n^ln(n) )

we get that

f(s) converges for Re(s) > 0.

Typical questions occur ; zero's , continuation to Re(s) < 0 , reflection ? natural boundary ?? etc

Im not sure if we have a summability method yet for Q = a_1 + a_2 + a_3 + ...

Can we just go around the pole at 0 , or are there issues ??

Does it have number theoretical or combinatorical meanings ?

regards

tommy1729

" Truth is what does not go away when you stop believing in it "

Tom Marcel Raes