MATHEMATICS BONUS FILES
for faculty and students
http://www2.onu.edu/~mcaragiu1/bonus_files.html
RECEIVED: November 1, 2007
PUBLISHED: November 7, 2007
The Riemann zeta function for undergraduates
Khristo N. Boyadzhiev
Department of Mathematics
Ohio Northern University
Ada, OH 45810
k-boyadzhiev@onu.edu
Abstract This is a short introduction to the zeta function including definition, integral
representations, functional equation, and statement of the Riemann hypothesis. The
functional equation of the Dirichlet L-function is also proved.
Contents.
1. Prerequisite
2. Definition of Zeta and integral representations
3. The functional equation
4. Functional equation of the Dirichlet function
5. Appendix
References
1. Prerequisite
We use only material from the standard single-variable Calculus course, basic facts about
complex variables, the definition of the Gamma function for
(1.1)
,
,
and from here, for any
,
(1.2)
.
Also the properties,
(1.3)
and when
,
(1.4)
.
An important item in the prerequisite is the integral (cosine transform of hyperbolic
cosecant)
(1.5)
,
which will be used to prove the functional equation os the zeta function. The best reference for
the beginner in this theory is [1] and then the excellent book of Titchmarsh [4]. An elementary
introduction based on classical analysis can be found in chapter 3 of [5].
2. Definition of Zeta and integral representations
It is well-known that the -series
1
(2.1)
converges for all
and defines a function
for such
. Using notations standard for this
theory
(2.2)
,
.
Consider also the “eta” function
(2.3)
,
then for
,
we have
, i.e.
(2.4)
, or
(2.5)
.
Equation (2.5) shows that the zeta function can be extended for all
.
Now we turn to some integral representations.
Proposition 1. For
and
,
(2.6)
,
(2.7)
,
(2.8)
,
2
(2.9)
,
(2.10)
.
Proof. It is clear that a proof is needed only for the case
(then use the substitution
).
Using geometric series
,
Exchanging the order of integration and summation we obtain in view of (1.2)
,
which proves (2.6) . Next, (2.7) is done in the same way. For (2.8) we write
.
Integrating term-wise we obtain
,
and (2.8) follows immediately, since
.
To prove (2.9) we write
in terms of exponentials and expand in geometric series.
3
.
Multiplying this by
and integrating from zero to infinity we obtain
,
where the right hand side is seen to be
. The representation (2.10) is proved
similarly.
Differentiating (2.9) and (2.10) for the variable
Corollary 1. For
we obtain two more representations.
and
(2.11)
,
(2.12)
.
It is good to notice that the integral in (2.11) is well defined for all complex values of
with
. We conclude that the following is true.
Corollary 2. The representation
(2.13)
(for any
) shows that
can be extended as analytic function for all
.
3. The functional equation
One of the most important properties of the zeta function is its functional equation
(3.1)
.
4
In view of (1.4) it can be written also as
(3.2)
.
We shall prove this equation using the idea of the “fourth method” in [4, p. 23]. First we need a
lemma.
Lemma. For
and every
(3.3)
.
This is a well-known Fresnel integral. Its evaluation will be given later in the Appendix.
Proposition 3. For every
Proof. Let first
the functional equation (3.2) holds.
. We shall evaluate the integral in (2.11) in a different way. Using (1.5)
and exchanging the order of integration we write
=
,
which in view of (2.8) equals
(3.4)
.
Comparing (3.4) to (2.11) we obtain the functional equation (3.2). The restriction
be dropped as both sides of (3.2) are well defined for
Proposition 4. The Riemann zeta function
all complex value
. At that,
can
.
can be extended as analytical function for
, and
.
5
Proof. The extension follows immediately from (2.13) and (3.2). According to (2.13), the zeta
function is well defined for
. The left hand side in (3.2) is well defined for
and hence the right hand side can be extended for all
also the zeros at
, as
,
. The functional equation shows
. Finally, setting
in (2.13) we
find the value of zeta at zero to be -1/2.
The zeros
are called trivial zeros.
The zeta function was studied by Leonhard Euler (1707-1783), who evaluated
for
.. In particular, he found the value
,
(see the next article in the Mathematics Bonus Files [2]). Fundamental contributions to the theory
of the zeta function were made by Bernhard Riemann (1826-1866), who stated the following
hypothesis in 1859.
The Riemann Hypothesis All nontrivial zeros of the zeta function are on the vertical line
.
The Riemann hypothesis is one of the most important open problems in mathematics. It has a
strong connection to the theory of prime numbers.
4. Functional equation of the Dirichlet function
It is good to see how the above technique can be used in other cases. For example, the
Dirichlet function
,
(4.1)
is similar to the Riemann zeta function. It is easy to show, exactly like in Proposition 1, that the
Dirichlet function has integral representation
6
.
Proposition 5. The Dirichlet function
(4.2)
satisfies the functional equation
,
(4.3)
or, equivalently, in view of (1.4),
.
(4.4)
Proof. This time we shall use the Fourier cosine transform
,
(4.5)
to evaluate the integral in (4.2) in a different way. For
.
Comparing this to (4.2) we obtain the desired functional equation (4.3).
The functional equation shows that
values of the variable
extends as analytical function for all complex
.
7
5. Appendix.
Evaluation of the Fresnel integral in (3.3). Let
and
. Using (1.2) after setting
we write
.
References
[1]
R. Ayoub, Euler and the Zeta Function. Amer. Math. Monthly, 81, 1067-1086, 1974.
[2]
K. N. Boyadzhiev, Euler’s formula for
: the Riemann zeta function and Bernoulli
numbers. Mathematics Bonus Files, 2007 (electronic)
[3]
H. M. Edwards,. Riemann's Zeta Function, Dover, New York, 2001.
[4]
E. C. Titchmarsh, The Theory of the Riemann Zeta Function, 2nd ed. Clarendon Press,
New York, 1987.
[5]
D.V. Widder, An Introduction to Transform Theory, Academic Press, New York, 1971.
8