Imaginary Zeros of Zeta Function

  • Thread starter rman144
  • Start date
  • #1
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I was doing some work with the zeta function and have a question.

I am aware that the Riemann Hypothesis claims that all of the critical zeros of the analytically continued zeta function have a real part Re(z)=1/2.

My question is, does the concept apply only to the complex zeros, or the imaginary and real parts separately.

Basically, is it possible to have:

Im(zeta(z))=0

Without having:

Re(zeta(z))=0


Or does a zero of one part automatically illustrate the existence of a zero for the other?
 

Answers and Replies

  • #2
107
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A zero x of a function f is when f(x)=0, (=0+0i) and it is no different with the zeta function.
 
  • #3
42
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Or does a zero of one part automatically illustrate the existence of a zero for the other?

No. For example, along the real line the imaginary part of the zeta function is zero, but the real part is certainly not always zero.

Look at this page on mathworld: http://mathworld.wolfram.com/RiemannZetaFunctionZeros.html
There's a graph of the curves where the real parts are zero and where the imaginary parts are zero. Where these curves intersect, that is, where the real and imaginary parts are zero, the function has a zero.
 
  • #4
18
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there is no exclusion for Im (z) in the hypothesis .Read again please.
 
  • #5
14
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Hi!
The Riemann Hypothesis clames that if RZF(z)=0 and z is not a trivial zero, then Re(z)=1/2. That is all. The real part of z needs to be equal to 1/2 (there is NOT restriccion about the imaginary part of z). And 0=0+0 I=ZERO.

RFZ= Riemann Zeta function.
 

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