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+--- Thread: Is it possible to extend the Euler product analytically? (/showthread.php?tid=558)
Is it possible to extend the Euler product analytically? - JmsNxn - 12/22/2010
I mean to say that an Euler product can be thought of as an iteration of multiplication and so therefore should have fractional iterates correct?
I think such an extension should probably obey, z E C, G is the gamma function, E(...) is an euler product: E( k=0, z ) k = G(z + 1)
I'm curious as to what
E(k=0, z) f(k) =?
Does anyone know where I could find something about this?
RE: Is it possible to extend the Euler product analytically? - tommy1729 - 12/23/2010
your question is not totally clear to me.
but afaik it is not known how everything can be put into a product.
for instance , i dont know an infinite product form that gives zeta(s) for all real parts with 1/2 < re(s) and NOT for re(s) < 1/2.
id love to see that.
a function can be analytically continued if it doesnt have a natural boundary , more specifically until it reaches a natural boundary.
and that is true independant of how the function is computed ( product , sum , integral , limit ) because the analytic computation form is another computation indepenent of how the original function was defined. ( since analytic continuation is unique ! )
hope i expressed myself clearly.
regards
tommy1729
RE: Is it possible to extend the Euler product analytically? - JmsNxn - 12/23/2010
Let me rephrase the question, considering the case for sigma
F(x) = sigma( k=0, x) f(k)
So therefore:
F(1) = f(0) + f(1)
F(2) = f(0) + f(1) + f(2)
I was just wondering how we could find rational and complex evaluations of F(x). The only requirement I believe it should have is:
F(x) + f(x+1) = F(x+1)
For the case of products
P(x) = E( k=0, x) f(k)
and therefore:
P(1) = f(0)*f(1)
P(2) = f(0)*f(1)*f(2)
etc etc
The only requirement P(x) requires is:
f(x+1)*P(x) = P(x+1) ; which as I was saying is not dissimilar to the gamma function. when f(x) = x, it is the gamma function plus one. For F(x), when f(x) = x, F(x) should equal Gauss' formula for sum of a series.
RE: Is it possible to extend the Euler product analytically? - bo198214 - 12/24/2010
This was discussed a lot on this board.
Search for keyword "continuum sum".
RE: Is it possible to extend the Euler product analytically? - tommy1729 - 12/24/2010
that solved the OP then.
but not my reply to it ...
i guess a nested solution is the only way :
find a function g(z) that converges for re > 1/2 only , find a product form f(z) for eta(z) that converges for re > a with a < 1/2.
then f(g(z)) is associated with the desired result.
now that i think of it , isnt there a product form known for eta(z) that works in the strip 0 < z < 2 ...
maybe there isnt a solution apart from the hadamard product form because log(eta(z)) is problematic because of the logaritmic riemann surface... (or leads us to hadamard anyway )
i guess the hadamard product form is the only solution for f(z) ??
i dont have time to consider this seriously at the moment ...
and it might not directly relate to tetration ... ( so sorry )
but i guess some are intrested in it anyway ?
or willing to help me
happy Xmas my fellow tetrationalists
tommy1729