The Riemann Hypothesis, proposed by Bernhard Riemann in 1859, is an unproven conjecture regarding the location of zeros of the Riemann zeta function. It states that all non-trivial zeros of the zeta function have a real part of 1/2. Proving or disproving the Riemann Hypothesis would have significant implications for mathematics and fields that rely on prime numbers like cryptography. It remains one of the most important unsolved problems in mathematics with a $1 million prize for a successful proof or disproof.

1. The Riemann Hypothesis
Dave Barnett, CISSP, CISM, CSDP, CSSLP
A brief discussion of the most important unproven hypothesis in mathematics. The
consensus of mathematicians is that the proof (or disproof) of the Riemann Hypothesis
will change the landscape of modern mathematics
Zeta functions: First -- Some Math Background
Functions
A “function” is like a mathematical machine: put a number in, do something to it, and out
pops another number. For example, let’s create a “doubler” function. We’ll call it “D”. If
we put in 2, out pops 4. If we put in 7, out pops 14. This function would be defined in
standard notation as:
D(a) = 2a
Where “a” is any number. (Both the “D” and the “a” are arbitrary labels – we could have
picked “X” and “g”, for example, or any letters in any language.) Spoken, this could be
“D of a is two a.”
Sum
The capital letter “S” in Greek (Σ, or Sigma ) is a standard shorthand symbol that means
“sum”. For example,
Σ (1,2,3,4) means 1+2+3+4
Often, there will be a number below and above the Sigma. This convention indicates that
you start with the number below, and go all the way up to the number on top, typically by
ones, adding each number in a running total. That is,
=Σ
10
1
1+2+3+4+5+6+7+8+9+10
The Greek letter Sigma is just a handy way of showing a particular pattern of addition.
This is especially true if it is used to represent an infinite series, which might otherwise
make your hand tired.
Product
The capital letter “P” in Greek (Π) is another standard shorthand symbol, which in this
case means “product”. For example,
© 2010 Dave Barnett Page 1

2. Π (1,2,3,4) means 1 x 2 x 3 x 4
Again, like Sigma, there may be a number below and above Π. This convention indicates
that you start with the number below, and go all the way up to the number on top,
typically by ones, adding each number in a running total. That is,
=Π
10
1
1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10
The Greek letter Π is a method of showing a particular pattern of multiplication.
Euler’s Zeta Function
In 1740, Leonhard Euler (pronounced ”Oiler”) came up with a function with interesting
properties that he called the zeta function. He used the lower case Greek letter “z” (ξ, or
zeta) to represent this function. It was defined as:
ξ(s) = 1/1s
+ 1/2s
+ 1/3s
+ 1/4s
… etc, all the way to infinity.
This infinite series can also be written as:
ξ(s) =
∞
=
Σ
1n n
s
1
The “s“ can be any number equal to or greater than one. (If it’s less than one, the function
has characteristics that aren’t interesting.) The n is the denominator, 1,2,3,4 etc., from 1
to infinity. The denominator n is raised to the power of s. If s is set to 2, this is:
....
36
1
25
1
16
1
9
1
2
1
1
1
etc++++++
Of course, mathematicians don’t really add an infinite series all the way to infinity. They
look for a formula that produces the same answer. In the case above, zeta of 2, or ξ(2), is
equal to π2
/6.
Euler discovered that ξ(s) = 1/1s
+ 1/2s
+ 1/3s
+ 1/4s
... from 1 to infinity is equivalent to:
....
13
1
1
1
11
1
1
1
7
1
1
1
5
1
1
1
3
1
1
1
.
2
1
1
1












−











−











−











−











−











− ssssss
Which can also be expressed as
p
s
1
1
1
− , in which p is a prime number from 2 to
infinity.
© 2010 Dave Barnett Page 2

3. The zeta function can also be written in mathematical notations as
ξ(s) = ∏
∞
−−
−
p
s
p 1
)1( =
∞
=
Σ
1n n
s
1
Riemann’s Zeta Function
In 1859, Riemann published a paper that used a variation of Euler’s zeta function to
investigate some of the characteristics of prime numbers. (He was working on a
conjecture by Gauss that the number of prime numbers smaller than n is approximately 1/
(logn(n)), or one over the natural log of n. (He didn’t succeed in a proof, but two other
mathematicians did in 1896. Gauss’ conjecture is known as the Prime Number Theorem.)
Riemann modified what s was allowed to be. Euler stated that s could be any integer
greater than or equal to 1. Riemann allowed s to be any complex number, as long as s did
not equal 1. A complex number is one that is made up of a Real number (e.g., -1, -2.3, 0,
1, 2 , 1.5, 2, 3, etc.) and i (the square root of negative 1). The format of a complex
number is a + bi, where a and b are Real numbers, such as 7 + 12i.
Running the zeta function on even negative numbers (e.g. –2, -4, -6, -8, etc) produces
zero as a result, a trivial solution. Aside from those zeros, Riemann proposed that his zeta
function had an infinite number of other solutions equaling 0s, all of the form ½ + ti,
where t is a real number, and i is the square root of –1. This conjecture, that all non-
trivial zeroes of zeta have a real part ½, is known as the Riemann hypothesis. It has been
shown to be true for the first 100 billion Real numbers t, but this isn’t considered a proof,
of course, There is currently a $1,000,000 prize waiting for anyone who can prove that
the Riemann hypothesis is true. With the proof of Fermat’s Last Theorem in 1996 by
Andrew Wiles, the Riemann hypothesis is now the most famous unsolved problem in
mathematics, and has been the most important for some time (at least 100 years.)
Why do we care?
Investigating the Riemann Hypothesis and zeta functions gives us information about the
patterns and density of prime numbers.
First of all, mathematicians care a great deal about this. Unlike Goldbach’s Conjecture
(any even number greater than two is the sum of two primes), which is merely
interesting, the Riemann Hypothesis has formed the foundation for a great deal of recent
work number theory. There are hundreds of papers written within the past 50 years that
begin with “assuming the Riemann Hypothesis is true.” If the Riemann Hypothesis is not
true, then a great deal of current number theory will have to be thrown out. That an
unproven hypothesis is used so frequently demonstrates some of its importance. But
that’s just for the mathematicians.
RSA public key cryptography is based on prime numbers. (It is really easy to multiply
two prime numbers, but very difficult to determine which numbers were used unless you
© 2010 Dave Barnett Page 3

4. already know one of them.) The Riemann Hypothesis may lead to the ability to detect
patterns in prime numbers. A deeper understanding of prime numbers may result in ways
to break the encryption used for Internet transactions (when you see the closed lock at the
bottom of your browser, you are using prime numbers to encrypt the data.) So, one of the
results of proving the Riemann Hypothesis may be that we can’t secure Internet traffic
using cryptography based on factorization.
Another implication of a proof of the Riemann is that it may result in a method for
solving an entire class of extremely difficult problems, known as NP-Complete
(http://en.wikipedia.org/wiki/NP-complete). NP-Complete problems are considered
“computationally infeasible.” That is, there is no algorithm for solving them. The typical
approach is to just keep guessing until you find the right answer. Once you have the right
answer, testing it is trivial. These include such problems as the factorization of the
product of large primes, The Traveling Salesman Problem, and shotgun sequencing of
Craig Venter’s genome.
Traveling Salesman Problem, and larIf a general method were developed to solve NP-
Complete problems, the impact on computer science, algorithms, engineering, and
science would be enormous. If any single NP-Complete problem can be solved, they all
can be quickly solved.
Another interesting aspect: Through a coincidence bordering on the miraculous,
Riemann’s zeta function was discovered to have a direct relationship to quantum systems.
Hugh Montgomery, who was working on the non-random pattern of the zeroes of the zeta
function was, by chance, introduced to Freeman Dyson at the Institute for Advanced
Study in Princeton. When Dyson politely asked what Montgomery was working on, and
Montgomery replied he had discovered an unusual distribution in the zeroes from
Riemann’s zeta. Dyson was astounded – the distribution was identical to that formed by
energy levels in a dynamic quantum system. Dyson, who was originally a mathematician
specializing in number theory, was arguably the only person in the world who had the
background to understand the math, and the knowledge of quantum physics to make the
connection. The chances of that particular conversation occurring between Dyson and
Montgomery were not very high.
This connection is now referred to as the Odlyzko-Montgomery Pair Correlation Law.
(http://www.maths.ex.ac.uk/~mwatkins/zeta/dyson.htm). The zeta function of the
Riemann Hypothesis is not only important to Number Theory, but also may describe
complex quantum systems.
If the Riemann Hypothesis is not true, we may have to throw away most of what we
understand about mathematics, from Pythagoras onward and start afresh.
David Hilbert would like to know the answer.
© 2010 Dave Barnett Page 4