The Gamma function has this -1 I don't understand
Yeah, this is completely historical. Edwards in his book on the Riemann zeta function tries to go back to using Gamma normalized in the obvious way so it agrees with factorial but that's never caught on.
In the case of the Riemann zeta function the key issue is that it is seen as more natural to have a plane of convegence for positive values of s. Moreover, writing it in that way, the values at positive integers are then natural and easy series.
cosine seems more primitive than sine
Can you expand on this one?
cosine seems more primitive than sine
It's a minor quibble. I think of cosine as the real part of e^ix, which is a very simple concept in my head. sine is the imaginary part of e^ix divided by i, which is slightly more complicated. If you had to relegate one to co- status, I'd choose sine.
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