Riemann zeta functionpole question?

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SUMMARY

The Riemann zeta function (RZF) exhibits a simple pole at s=1, attributed to its divergence to positive infinity, as demonstrated by the harmonic series 1 + 1/2 + 1/3 + ... which sums to infinity. The discussion highlights that while the RZF is positive at s=1, it takes on negative values for inputs approaching 1/2, raising questions about the existence of zeros along the real line. Specifically, the inquiry focuses on the transition from positive to negative values without crossing zero, indicating a potential missing zero in the vicinity of the pole.

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lostcauses10x
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The simple pole at on is due to that its value of course is not closed due to it is an infinite value.

My question is: is this value of infinity, positive or negative. or both??
 
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positive. The RZF has a pole at s=1 because at s=1 it is equal to the harmonic series sum 1+1/2+1/3+1/4+1/5+...= infinity
 
camilus said:
positive. The RZF has a pole at s=1 because at s=1 it is equal to the harmonic series sum 1+1/2+1/3+1/4+1/5+...= infinity

Thank you:

So next question is,

Since say at 1/2 the value of the zeta function is negative. and tends in the negative direction continuing closer you get to one (of course with the limit less than one):and the pole at one is positive, were along the real line is a missing zero?

Positive value going to a negative value on a line due to a function and no cross over in value at zero ?
 
Last edited:

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