From p.1 of Connes' "Trace formula in noncommutative geometry and the zeros
of the Riemann zeta function"

"We shall give in this paper a spectral interpretation of the zeros
of the Riemann zeta function...The spectral interpretation of the zeros
of zeta will be as an absorption spectrum, *i.e.* as missing
spectral lines. **All zeros** will play a role in the spectral side
of the trace formula, but while the critical zeros will appear *per
se*, the noncritical ones will appear as resonances and enter in the
trace formula through their harmonic potential with respect to the
critical line."

]:
"The winner could well be Alain Connes, a mathematician based at the
Institute of Advanced Scientific Study in Bures-sur-Yvette, France.
Connes has a startlingly direct approach to the problem: create a
system that already includes the prime numbers. To understand how, you
have to imagine a quantum system not as a particle bouncing around an
atom, say, but as a geometrical space. It sounds odd, but it represents
one of the weird things about quantum systems: they can be two or more
things at once.

Like Schrödinger's cat, which is a peculiar mixture of dead
and alive, any quantum object can find itself in a "superposition" of
different states. To characterise this messy existence, physicists use
what they call a state space. For each kind of possibility (say
"alive" and "dead"), you draw a new axis and add a dimension to the
space. If there are just two possible states, as is the case for
Schrödinger's cat, the space is two dimensional, with three
states it is three dimensional, and so on.

Then in the Schrödinger's cat space, you would mark a cross
one unit along the *x*-axis to represent a fully alive cat.
Similarly, a stone dead cat would be one unit up the *y*-axis,
and a part-alive, part-dead cat would appear somewhere along an arc
between these points.

The "shape" of the space affects how the state moves around in it,
and therefore how the system works, including the way its energy levels
are arrayed. This depends not just on the number of dimensions, but
also on the geometry of how they are stuck together.

Connes decided to build a quantum state space out of the prime
numbers. Of course, the primes are a bunch of isolated numbers,
nothing like the smooth expanses of space in which we can measure
things like angles and lengths. But mathematicians have invented some
bizarrely twisted geometries that are based on the primes. In "5-adic"
geometry, for example, numbers far apart (in the ordinary way) are
pulled close together if they differ by 5, or 15, or 250--any multiple
of 5. In the same way, 2-adic geometry pulls together all the even
numbers.

To put all the primes in the mix, Connes constructed an
infinite-dimensional space called the Adeles. In the first dimension,
measurements are made with 2-adic geometry, in the second dimension
with 3-adic geometry, in the third dimension with 5-adic geometry, and
prime numbers.

Last year Connes proved that his prime-based quantum system has
energy levels corresponding to all the Riemann zeros that lie on the
critical line. He will win the fame and the million-dollar prize if he
can prove that there aren't any extra zeros hanging around, unaccounted
for by his energy levels.

That last step is a formidable one. Has Connes simply replaced the
Riemann hypothesis with an equally difficult question? Some experts
advise caution. "I still think that some major new idea is needed here,"
says Bombieri."