Is zeta(-1) equal to -1/12? A Discussion on the Infamous Sum of Natural Numbers

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In summary, the conversation discusses the concept of zeta(x), a function that can be extended to values of x smaller than 1. This extension allows for calculations such as zeta(-1) = -1/12, which has caused confusion regarding the sum of natural numbers. The conversation provides a link to a thread discussing this topic in more detail.
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Hi,
I have read that zeta(x) = 1^(-x) + 2^(-x) + 3^(-x) + ... infinity
for x = -1, zeta(-1) = 1 + 2 + 3 + 4 + 5 ...
What confused me is that zeta(-1) = -1/12 and so 1 + 2 + 3 + 4 + 5 + ... = -1/12
Can anybody give a proof that zeta(-1) is -1/12.

Thanks in advance.
 
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We have a fairly substantial thread on this topic already

https://www.physicsforums.com/showthread.php?t=732197

The main gist is that [itex] \zeta(x) [/itex] is a function such that if x>1,
[tex] \zeta(x) = 1^{-x} + 2^{-x} +... [/tex]
You can extend this function to allow for values of x smaller than 1, but when you do you aren't really calculating the sum of natural numbers anymore. In particular trying to do manipulations as if you really are calculating that sum can be dangerous. See the thread for more details and feel free to continue the discussion there.
 
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FAQ: Is zeta(-1) equal to -1/12? A Discussion on the Infamous Sum of Natural Numbers

What is the proof that zeta(-1) is equal to -1/12?

The proof involves using analytic continuation of the Riemann zeta function, which is defined as ∑n=1∞ 1/n^s for s>1. By extending the definition to s=-1, we can use a regularization technique to show that the value of zeta(-1) is equal to -1/12.

How does the analytic continuation of the Riemann zeta function work?

The analytic continuation of the Riemann zeta function involves extending the function's domain beyond its original definition. This allows us to use complex numbers and other techniques to evaluate the function at different points, including s=-1, which gives us the value of zeta(-1) as -1/12.

Is -1/12 the only possible value for zeta(-1)?

No, -1/12 is not the only possible value for zeta(-1). In fact, zeta(-1) is not defined at s=-1, so we cannot assign a single value to it. However, using the analytic continuation method, we can find a value for zeta(-1) that satisfies certain mathematical properties and is consistent with other calculations.

What are some applications of the proof that zeta(-1) is equal to -1/12?

One major application is in theoretical physics, specifically in quantum field theory. The value of zeta(-1) is related to the number of possible energy levels of a quantum system, so the proof has implications for understanding the behavior of particles at the smallest scales.

Is there any controversy surrounding the proof that zeta(-1) is equal to -1/12?

Yes, there is some controversy surrounding the proof. While the proof is mathematically rigorous and has been accepted by most mathematicians and physicists, some argue that the regularization technique used to evaluate zeta(-1) is not valid. Others argue that the value of zeta(-1) as -1/12 is purely a mathematical construct and has no physical significance. However, the proof continues to be studied and used in various fields of mathematics and physics.

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