- #1
zaybu
- 53
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In String Theory, one of the most crucial calculations involves summing all the numbers from 1 to infinity, which obviously should be infinite. But not in ST, where the Rieman Zeta Function is used, and gives a value of -1/12?!
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.
If one uses the functional theorem:
ζ(s)= 2s π s-1 sin(πs/2) ζ(1-s)Γ(1-s), where Γ is the well-known gamma function.
And let s = -1
ζ(-1)= 2-1 π -2 sin(-π/2) ζ(2)Γ(2)
ζ(-1)= (1/2) (1/π2)(-1) (π2/6)(1) = -1/12
This yields the dreaded sum:
ζ(-1) = Σ 1/n-1 = Σ n = 1 + 2 + 3 +... = -1/12
What I find strange is that this calculation is done with complex numbers on a complex plane, using analytic continuation. But how does one explain that the initial sum deals with real numbers, not complex numbers? This result is counter common sense. Any help would be greatly appreciated.
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.
If one uses the functional theorem:
ζ(s)= 2s π s-1 sin(πs/2) ζ(1-s)Γ(1-s), where Γ is the well-known gamma function.
And let s = -1
ζ(-1)= 2-1 π -2 sin(-π/2) ζ(2)Γ(2)
ζ(-1)= (1/2) (1/π2)(-1) (π2/6)(1) = -1/12
This yields the dreaded sum:
ζ(-1) = Σ 1/n-1 = Σ n = 1 + 2 + 3 +... = -1/12
What I find strange is that this calculation is done with complex numbers on a complex plane, using analytic continuation. But how does one explain that the initial sum deals with real numbers, not complex numbers? This result is counter common sense. Any help would be greatly appreciated.