|The real part (red) and imaginary part (blue) of the critical line Re(s) = 1/2 of the Riemann zeta-function.
The Riemann 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.
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 Riemann hypothesis is one of the most important open problems in 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
Klein bottle is an example of a closed
surface (a two-dimensional
manifold) that is
non-orientable (no distinction between the "inside" and "outside"). This image is a representation of the object in everyday three-dimensional space, but a true Klein bottle is an object in
four-dimensional space. When it is
constructed in three-dimensions, the "inner neck" of the bottle curves outward and intersects the side; in four dimensions, there is no such self-intersection (the effect is similar to a
two-dimensional representation of a cube, in which the edges seem to intersect each other between the corners, whereas no such intersection occurs in a true
three-dimensional cube). Also, while any real, physical object would have a thickness to it, the surface of a true Klein bottle has no thickness. Thus in three dimensions there is an inside and outside in a colloquial sense: liquid forced through the opening on the right side of the object would collect at the bottom and be contained on the inside of the object. However, on the four-dimensional object there is no inside and outside in the way that a
sphere has an inside and outside: an unbroken curve can be drawn from a point on the "outer" surface (say, the object's lowest point) to the right, past the "lip" to the "inside" of the narrow "neck", around to the "inner" surface of the "body" of the bottle, then around on the "outer" surface of the narrow "neck", up past the "seam" separating the inside and outside (which, as mentioned before, does not exist on the true 4-D object), then around on the "outer" surface of the body back to the starting point (see the light gray curve on
this simplified diagram). In this regard, the Klein bottle is a higher-dimensional analog of the
Möbius strip, a two-dimensional manifold that is non-orientable in ordinary 3-dimensional space. In fact, a Klein bottle
can be constructed (conceptually) by "gluing" the edges of two Möbius strips together.