bezdomni
27th September 2006, 07:03
n mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians. Unlike some other celebrated problems, it is more attractive to professionals in the field than to amateurs.
The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta-function ζ(s). The Riemann zeta-function is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s = −2, s = −4, s = −6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:
The real part of any non-trivial zero of the Riemann zeta function is ½.
Thus the non-trivial zeros should lie on the so-called critical line ½ + it with t a real number and i the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line.
The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zeta-function. You can see the first non-trivial zeros at Im(s) = ±14.135, ±21.022 and ±25.011.
Enlarge
The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zeta-function. You can see the first non-trivial zeros at Im(s) = ±14.135, ±21.022 and ±25.011.
A polar graph of zeta, that is, Re(zeta) vs. Im(zeta), along the critical line s=it+1/2, with t running from 0 to 34
Enlarge
A polar graph of zeta, that is, Re(zeta) vs. Im(zeta), along the critical line s=it+1/2, with t running from 0 to 34
The Riemann hypothesis is one of the most important open problems of contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned, from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class.)
From Wikipedia (http://en.wikipedia.org/wiki/Riemann_hypothesis)
Can anybody explain this is more understandable terms? I understand the zeta function, but I know there is some of it that I'm not getting....considering that most of the terminology is over my head.
I am really fascinated by the unsolved math problems.
The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta-function ζ(s). The Riemann zeta-function is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s = −2, s = −4, s = −6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:
The real part of any non-trivial zero of the Riemann zeta function is ½.
Thus the non-trivial zeros should lie on the so-called critical line ½ + it with t a real number and i the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line.
The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zeta-function. You can see the first non-trivial zeros at Im(s) = ±14.135, ±21.022 and ±25.011.
Enlarge
The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zeta-function. You can see the first non-trivial zeros at Im(s) = ±14.135, ±21.022 and ±25.011.
A polar graph of zeta, that is, Re(zeta) vs. Im(zeta), along the critical line s=it+1/2, with t running from 0 to 34
Enlarge
A polar graph of zeta, that is, Re(zeta) vs. Im(zeta), along the critical line s=it+1/2, with t running from 0 to 34
The Riemann hypothesis is one of the most important open problems of contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. (J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned, from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class.)
From Wikipedia (http://en.wikipedia.org/wiki/Riemann_hypothesis)
Can anybody explain this is more understandable terms? I understand the zeta function, but I know there is some of it that I'm not getting....considering that most of the terminology is over my head.
I am really fascinated by the unsolved math problems.