Riemann Zeta Function - Zeros, The Critical Line, and The Riemann Hypothesis

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 {sC : 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 {sC : 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:

    The male has been persuaded to assume a certain onerous and disagreeable rôle with the promise of rewards—material and psychological. Women may in the first place even have put it into his head. BE A MAN! may have been, metaphorically, what Eve uttered at the critical moment in the Garden of Eden.
    Wyndham Lewis (1882–1957)

    Oversimplified, Mercier’s Hypothesis would run like this: “Wit is always absurd and true, humor absurd and untrue.”
    Vivian Mercier (b. 1919)