If you know about complex numbers, you will be
able to appreciate one of the great unsolved problems
of our time. The Riemann zeta function is defined by
Zeta(z) = SUM_{k=1 to infinity}
(1/k^{z}) .
This is the harmonic series for z=1 and
Sums of Reciprocal Powers if you set z equal to
other positive integers. The function can be extended to
the entire complex plane (with some poles) by a process
called "analytic continuation", although what that is
won't concern us here.
It is of great interest to find the zeroes of this function.
The function is trivially zero at the negative even
integers, but where are all the other zeroes? To date,
the only other zeroes known all lie on the line
in the complex plane with real part
equal to 1/2. This has been checked for several hundred
million zeroes!
No one knows, however, if all of the
infinite number of nontrivial zeroes lie on
this line; the conjecture that they do is called the
Riemann hypothesis and is one of the great unsolved
problems of mathematics, dating back to 1859.
Presentation Suggestions:
Even though students may not understand all
the technical details of this Fun Fact,
they usually find it fascinating that
a great unsolved problem amounts to finding all the zeroes
of some function (a concept they are familiar with) and that
it turns out to be related to other things that they may
have heard about.
The Math Behind the Fact:
Many other problems in number theory,
such as ones involving the distribution of primes,
have been shown to be related to the Riemann hypothesis,
so answering this would provide insight to a whole bunch
of other problems! For instance,
the Prime Number Theorem gives a
good approximation to how many primes are less than a
given number, but the Riemann hypothesis is related to
a conjecture about how good that approximation is!
To see why prime numbers relate to zeta functions,
see Euler's Identity. Another connection
that mathematicians are currently exploring
is why the spacings of
zeroes of the Riemann zeta function resemble the
spacings of eigenvalues of random unitary matrices.
How to Cite this Page:
Su, Francis E., et al. "Riemann Hypothesis."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
