Friday, 22 March 2019

number theory - Explanation of Zeta function and why 1+2+3+4+... = -1/12











I found this article on Wikipedia which claims that n=0n=1/12. Can anyone give a simple and short summary on the Zeta function (never heard of it before) and why this odd result is true?


Answer



The answer is much more complicated than limx0sin(x)x.



The idea is that the series n=11nz it is convergent when Re(z)>1, and this works also for complex numbers.



The limit is a nice function (analytic) and can be extended in an unique way to a nice function ζ. This means that




ζ(z)=n=11nz;Re(z)>1.



Now, when z=1, the right side is NOT convergent, still ζ(1)=112. Since ζ is the ONLY way to extend n=11nz to z=1, it means that in some sense



n=11n1=112



and this is exactly what that means. Note that, in order for this to make sense, on the LHS we don't have convergence of series, we have a much more suttle type of convergence: we actually ask that the function n=11nz is differentiable as a function in z and make z1...



In some sense, the phenomena is close to the following:




n=0xn=11x;|x|<1.



Now, the LHS is not convergent for x=2, but the RHS function makes sense at x=2. One could say that this means that in some sense n=02n=1.



Anyhow, because of the Analyticity of the Riemann zeta function, the statement about ζ(1) is actually much more suttle and true on a more formal level than this geometric statement...


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