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Taylor-Dirichlet conversion
#1
I just found a simple way to convert between a Taylor series at 0 and a Dirichlet series.
This requires a fractional differintegral.

For any function with :



Pretty cool, huh?

Using this conversion, the Riemann Zeta function can be expressed as:
which I believe is somewhat well known due to its connection with Bernoulli polynomials, and 1/(1 - e^z) is well known to be the generating function for Bernoulli polynomials. Still, I've never seen this formula before, it is interesting to see it like this.
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#2
Hmm. If the above is true, then
(which is false)
but

is the actual result.

So I must have made a slight mistake... I'll have to think on this.

However, a similar form gives:

because
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