Trivial zeros of the Riemann zeta function

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Discussion Overview

The discussion revolves around the trivial zeros of the Riemann zeta function, particularly focusing on the confusion regarding the evaluation of the zeta function at negative even integers. Participants explore the definitions and properties of the zeta function in different domains.

Discussion Character

  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses confusion about how negative even numbers can be zeros of the Riemann zeta function, citing a specific evaluation that appears non-zero.
  • Another participant points out that the initial definition used is valid only for values where the real part is greater than 1, implying a misunderstanding of the domain.
  • A further reply clarifies that the analytic continuation of the zeta function is necessary for evaluating it at negative integers.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the initial confusion, but there is agreement on the need for analytic continuation to understand the behavior of the zeta function at negative even integers.

Contextual Notes

The discussion highlights limitations in understanding the zeta function's definition and its analytic properties, particularly regarding the domain of validity for its initial definition.

mrbohn1
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Clearly I am missing something obvious here, but how is it that negative even numbers are zeros of the Riemann zeta function?

For example:

\zeta (-2)=1+\frac{1}{2^{-2}}+\frac{1}{3^{-2}}+...=1+4+9+..

Which is clearly not zero. What is it that I am doing wrong?
 
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You are using the definition of zeta(s) for Re(s) > 1 with a number that has a real part smaller than or equal to 1.
 
mrbohn1 said:
What is it that I am doing wrong?

You need not the function you posted, but its analytic continuation.
 
Thanks! It all becomes clear.
 

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