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Certianly there is a lot of reference material on series transformations: they accelerate convergence, provide analytic continuations and what not. But I have not yet seen a like presentation of product transformations. Given that there are ways to write a product as a series, and vice-versa (see this post), would it be so difficult to translate known series transformations into product transformations?
An example application would be the Riemann zeta fcn, the series definition can be analytically continued to the whole complex plane (except z=1) via some clever series manipulation + a series transformation (see this post), but has anybody ever used similar techniques to analytically continue the Euler product over primes representation of the Riemann zeta? Yeah, I know about the Hadamard Product derived using the Weierstrass formula, but that is not what I'm after.
Just fishing for your ideas,
-Ben
An example application would be the Riemann zeta fcn, the series definition can be analytically continued to the whole complex plane (except z=1) via some clever series manipulation + a series transformation (see this post), but has anybody ever used similar techniques to analytically continue the Euler product over primes representation of the Riemann zeta? Yeah, I know about the Hadamard Product derived using the Weierstrass formula, but that is not what I'm after.
Just fishing for your ideas,
-Ben
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