The zeta Function, The Riemann Hypothesis
The Riemann Hypothesis
If you have been sent here by a search engine,
I should warn you that this is a very complicated topic.
A high school student can understand Fermat's Last Theorem (as a problem in number theory),
but one needs a substantial background in convergent series and complex analysis
to understand the statement of the riemann hypothesis,
and even more background is required to understand its significance
in algebraic and analytic number theory.
If you are not familiar with the zeta function on the complex plane,
please go back to the beginning of this topic.
Assume ζ(s) is defined on the right half plane, as described in the previous section.
The riemann hypothesis, abbreviated rh,
states that ζ(s) is never zero, except for points along the vertical line re(s) = ½.
The generalized riemann hypothesis, abbreviated grh, makes the same claim about every
lχ,m series.
Other topics on this website will explore the ramifications of rh and grh.
For now, the zeta function, and its variations, have been defined,
and rh has been presented as an open conjecture,
widely believed, but still unproven.