Some refresher:

The zeta function is given by ζ(s) = Σ 1/n

^{s}, where s is any complex number. The earliest calculation of this function was made by Euler with s = 2

ζ(2) = Σ1/n

^{2}= 1/1

^{2}+ 1/2

^{2}+ 1/3

^{2}+ ... = π

^{2}/6.

Note: for s = -1

ζ(-1) = 1 + 2 + 3 +... = -1/12

This is what prodded me to investigate this sum, as mentioned in Is String Theory wrong?

In my investigation I came across the functional theorem:

ζ(s)= 2

^{s}π

^{s-1}sin(πs/2) ζ(1-s)Γ(1-s), where Γ is the well-known gamma function.

So if we let s = -1

ζ(-1)= 2

^{-1}π

^{-2}sin(-π/2) ζ(2)Γ(2)

ζ(-1)= (1/2) (1/π

^{2})(-1) (π

^{2}/6)(1) = -1/12

This yields the dreaded sum:

ζ(-1) = Σ 1/n

^{-1}= Σ n = 1 + 2 + 3 +... = -1/12

There are other ways far too long to get this result, this being the shortest one. So I ask you to forgive my indulgence.

So what can I conclude? After reviewing complex analytic functions, the Cauchy-Riemann equations, complex integration and Cauchy's integral theorem, power series, the residue theorem and analytic continuation, I can only say that the mathematics is consistent.

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