Riemann-Zeta Function Analytical Continuation + Sorcery Watch

Astronomical
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I've been doing a little bit of reading about string theory (bosonic, fyi) and in one of the derivations it was just pulled out of a hat that the sum of all the integers is
\displaystyle \sum_{n=1}^{\infty} n = -\frac{1}{12}.

Now, obviously this just looked wrong to me, so I went a little further into my reading and it turns out this result is actually the evaluation of \zeta(-1), i.e. the Riemann-Zeta function, defined by
\displaystyle \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

To my utter dismay, I then found out that this function has trivial zeros at all the even negative integers, i.e.
\zeta(-2) = \zeta(-4) = \zeta(-6) = ... = \zeta(-2n) = 0,\ \forall n \in \mathbb{N}

That is to say that
\zeta(-1) = 1 + 2 + 3 + 4 + 5 + ... = -1/12

\zeta(-2n) = 1 + 2^{2n} + 3^{2n} + 4^{2n} + 5^{2n} + ... = 0

Apparently this ludicrous behaviour follows from the "analytic continuation" of the original definition of the zeta function to the complex plane. I've seen Riemann's paper on this, but it was obviously in German and I didn't understand it. The analytically continued version is
\displaystyle \zeta(s) = 2^s \pi^{s-1} \sin(\pi s/2) \Gamma(1-s) \zeta(1-s),\ s=\sigma + i \tau \in \mathbb{C}

Can anyone please explain to me how he arrived at this equation? I understand that it allows you to extend the domain by relating the value at s to the value at 1-s but how do you arrive there? And how can it possibly be consistent if it produces the above results? :eek:
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davros
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(Original post by Astronomical)
I've been doing a little bit of reading about string theory (bosonic, fyi) and in one of the derivations it was just pulled out of a hat that the sum of all the integers is
\displaystyle \sum_{n=1}^{\infty} n = -\frac{1}{12}.

Now, obviously this just looked wrong to me, so I went a little further into my reading and it turns out this result is actually the evaluation of \zeta(-1), i.e. the Riemann-Zeta function, defined by
\displaystyle \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}

To my utter dismay, I then found out that this function has trivial zeros at all the even negative integers, i.e.
\zeta(-2) = \zeta(-4) = \zeta(-6) = ... = \zeta(-2n) = 0,\ \forall n \in \mathbb{N}

That is to say that
\zeta(-1) = 1 + 2 + 3 + 4 + 5 + ... = -1/12

\zeta(-2n) = 1 + 2^{2n} + 3^{2n} + 4^{2n} + 5^{2n} + ... = 0

Apparently this ludicrous behaviour follows from the "analytic continuation" of the original definition of the zeta function to the complex plane. I've seen Riemann's paper on this, but it was obviously in German and I didn't understand it. The analytically continued version is
\displaystyle \zeta(s) = 2^s \pi^{s-1} \sin(\pi s/2) \Gamma(1-s) \zeta(1-s),\ s=\sigma + i \tau \in \mathbb{C}

Can anyone please explain to me how he arrived at this equation? I understand that it allows you to extend the domain by relating the value at s to the value at 1-s but how do you arrive there? And how can it possibly be consistent if it produces the above results? :eek:
Think about the series 1 + z + z^2 + z^3 + .... This series converges for |z| < 1 and within that range it converges to \dfrac{1}{1 - z} so we can extend the original series to a function that is defined everywhere apart from z = 1.

But that doesn't mean you can stick z = 2 into the series and say that 1 + 2 + 2^2 + ... = \dfrac{1}{1-2} = -1. All you're doing is making an inappropriate formal substitution into a series that doesn't converge!

There's a nice derivation of Riemann's functional equation for the zeta function in an appendix of Julian Havil's excellent little book 'Gamma', and you should find a more formal treatment in most decent Complex Analysis books
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