Zeros, The Critical Line, and The Riemann Hypothesis
The functional equation shows that the Riemann zeta function has zeros at −2, −4, ... . These are called the trivial zeros. They are trivial in the sense that their existence is relatively easy to prove, for example, from sin(πs/2) being 0 in the functional equation. The non-trivial zeros have captured far more attention because their distribution not only is far less understood but, more importantly, their study yields impressive results concerning prime numbers and related objects in number theory. It is known that any non-trivial zero lies in the open strip {s ∈ C : 0 < Re(s) < 1}, which is called the critical strip. The Riemann hypothesis, considered one of the greatest unsolved problems in mathematics, asserts that any non-trivial zero s has Re(s) = 1/2. In the theory of the Riemann zeta function, the set {s ∈ C : Re(s) = 1/2} is called the critical line. For the Riemann zeta function on the critical line, see Z-function.
Read more about this topic: Riemann Zeta Function
Famous quotes containing the words critical and/or hypothesis:
“It is a sign of our times, conspicuous to the coarsest observer, that many intelligent and religious persons withdraw themselves from the common labors and competitions of the market and the caucus, and betake themselves to a certain solitary and critical way of living, from which no solid fruit has yet appeared to justify their separation.”
—Ralph Waldo Emerson (18031882)
“The hypothesis I wish to advance is that ... the language of morality is in ... grave disorder.... What we possess, if this is true, are the fragments of a conceptual scheme, parts of which now lack those contexts from which their significance derived. We possess indeed simulacra of morality, we continue to use many of the key expressions. But we havevery largely if not entirelylost our comprehension, both theoretical and practical, of morality.”
—Alasdair Chalmers MacIntyre (b. 1929)