Thursday, March 31, 2011

Zeta function is ok

This is a sequel to, Is String Theory wrong?.

Some refresher:

The zeta function is given by ζ(s) = Σ 1/ns, where s is any complex number. The earliest calculation of this function was made by Euler with s = 2

ζ(2) = Σ1/n2 = 1/12 + 1/22 + 1/32+ ... = π2/6.

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)= 2s π s-1sin(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 -2sin(-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.