Complex Zeros in Riemann Zeta Function: Is it Possible?

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SUMMARY

The discussion centers on the possibility of finding two complex zeros of the Riemann Zeta function that share the same imaginary part but have different real parts, specifically outside the critical strip. Participants debate the implications of such pairs on the Riemann Hypothesis (RH), which asserts that all nontrivial zeros lie on the critical line. The conversation highlights the lack of definitive proof regarding the existence of such pairs and the ongoing efforts in the mathematical community to explore these concepts further.

PREREQUISITES
  • Understanding of the Riemann Zeta function and its properties
  • Familiarity with the Riemann Hypothesis and its implications
  • Knowledge of complex analysis, particularly regarding zeros of analytic functions
  • Basic understanding of functional equations in mathematics
NEXT STEPS
  • Research the implications of the Riemann Hypothesis on complex zeros of the Zeta function
  • Study the functional equation of the Riemann Zeta function, specifically \(\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s)\)
  • Explore existing proofs related to the distribution of nontrivial zeros of the Riemann Zeta function
  • Investigate the concept of trivial zeros and their role in the context of the Riemann Zeta function
USEFUL FOR

Mathematicians, number theorists, and students interested in the complexities of the Riemann Zeta function and the Riemann Hypothesis.

  • #31
Your math page requires a password and username, I am very skeptical as to joining a commercial site, especially if I'm unsure of the content.

When you say zeta(s-1/2) on the critical strip, do you mean when 1 > re s > 0 or when 1 > s-1/2 > 0 ?

Also if you show that zeta(s) is a zero in the critical strip, then you've already shown that zeta(1-s) is one.

If your proof is so simple, you should write it here.
 
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  • #32
solamon please, and I'm not saying it because I'm bad, and i don't want to hurt anyone, but please- SHUT UP!

p.s. please lock this thread.
 

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