Great answer, thanks.
Yes, my shallow, uninformed by higher maths complaint about the zeta function is that it sums n^-s instead of the simpler n^s.
As I said, the Riemann zeta function has its definition because it makes sense and the other doesn't. Once you have a solid definition of ζ(-1), you could declare that 1+2+3+...=ζ(-1) and then you might be tempted to reverse the sign. But the zeta function was around for a century before Riemann encouraged people to emphasize the values that don't make immediate sense.
You can do an awful lot just having it defined for real s>1. Euler used it to prove the infinitude of primes: ζ(1) is the harmonic series, thus infinite (or more precisely, an infinite li...
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