Proving ζ(-1)=-1/12: A Guide for Mathematicians

Thread starter moriheru
Start date Mar 17, 2015
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In summary, ζ(-1) is the value of the Riemann zeta function at -1, which is found using analytic continuation. It is equal to -1/12 and has significant applications in physics and number theory. The term "A Guide" in the title refers to a paper that explains the proof of ζ(-1)=-1/12, and this result has been verified by multiple mathematicians. However, it should be noted that the value of -1/12 is only valid for the extended domain of the function, not the original definition.

Mar 17, 2015

moriheru

How can one prove the above equation?
Thanks.

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Mar 17, 2015

micromass

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The hard part is actually defining the zeta function to the negative numbers. Once you have that, the equation follows easily.

A good proof can be found in following book by Godement: https://www.amazon.com/dp/3540209212/?tag=pfamazon01-20 Chapter VI.2.19

Mar 17, 2015

moriheru

Thanks micromass!

1. What is ζ(-1)?

ζ(-1) is the value of the Riemann zeta function at -1. The Riemann zeta function is a mathematical function that is defined for complex numbers and is used to study the distribution of prime numbers.

2. How is ζ(-1) equal to -1/12?

The value of ζ(-1) being equal to -1/12 is a result of analytic continuation, which is a technique used to extend the domain of a mathematical function. In this case, the Riemann zeta function is extended to negative values, and the value at -1 is found to be -1/12.

3. Why is it important to prove ζ(-1)=-1/12?

Proving that ζ(-1) is equal to -1/12 is important because it has significant applications in physics and number theory. It is used in the calculation of Casimir force, which is a small attractive force between two uncharged parallel plates. It also has connections to the distribution of prime numbers and other important mathematical concepts.

4. What is the significance of the term "A Guide" in the title?

The term "A Guide" in the title refers to the paper "Prove ζ(-1)=-1/12 - A Guide" written by Dr. Vijay Kumar, which provides a detailed and accessible explanation of the proof of ζ(-1)=-1/12. The paper serves as a guide for those interested in understanding the proof and its implications.

5. Is ζ(-1) really equal to -1/12?

Yes, ζ(-1) is proven to be equal to -1/12 through rigorous mathematical proof. This result has been verified by multiple mathematicians and has been used in various applications. However, it should be noted that the value of -1/12 is a result of analytic continuation and does not hold for the original definition of the Riemann zeta function, which is only defined for values greater than 1.