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 Is it possible to extend the Euler product analytically? JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 12/22/2010, 08:50 PM (This post was last modified: 12/22/2010, 09:29 PM by JmsNxn.) 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? tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 12/23/2010, 01:29 PM 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 JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 12/23/2010, 10:18 PM (This post was last modified: 12/23/2010, 10:27 PM by JmsNxn.) 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. bo198214 Administrator Posts: 1,395 Threads: 91 Joined: Aug 2007 12/24/2010, 02:11 AM This was discussed a lot on this board. Search for keyword "continuum sum". tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 12/24/2010, 01:15 PM 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 « Next Oldest | Next Newest »

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