1. Introduction
If you do not know about the Riemann zeta function, then do an internet search to observe the extensive research that has been done investigating various properties of this function. A more detailed introduction to the Riemann zeta function can be found in the references [1] [2]. One way of defining this function is to express it as an infinite series having the form
(1)
where
is a real number greater than 1 in order for the infinite series to converge. Observing that for
, the series becomes the harmonic series which slowly diverges. The zeta function was introduced by Leonhard Euler (1707-1783) who considered
to be a function of a real variable.
Another form for representing the zeta function is the integral representation
(2)
where
is the gamma function.
Bernhard Riemann (1826-1866) studied the zeta function and changed the independent real variable
to the complex variable
. This notation is still used in current studies of the zeta function. By doing this, Riemann made
a function of a complex variable. Riemann discovered that the zeta function satisfied the functional equation
(3)
where
is the gamma function. Several proofs of the above result can be found in the Titchmarsh reference [3]. Various forms for the functional equation are derived later in this paper. The equation allowed the zeta function to be defined for values
. The point
is a singular point. Using properties of the gamma function, the functional equation can be expressed in the alternative form
(4)
derived later in this paper. The above results can be used to extend the definition of the zeta function to the whole of the complex plane.
Euler also showed that the zeta function can also be expressed using prime numbers
(5)
where the product runs through all primes
. The equation (5) is known as the Euler product formula.
The Euler-Riemann function
is an important function in number theory where it is related to the distribution of prime numbers. It also can be found in such diverse study areas as probability and statistics, physics, Diophantine equations, modular forms and in many tables of integrals. The Euler-Riemann zeta function evaluated at special integer values for s occurs quite frequently in tables of integrals and in many areas of science and engineering.
2. Bernoulli and Euler Numbers
In later sections we need knowledge of the Bernoulli numbers
and Euler numbers
. Representation of these numbers can be obtained from reference [4] (24.2), where one finds the generating functions
(6)
Note that the first few values are
Note that for
positive integers with
,
and for
,
.
3. Calculation of
for
Leonhard Euler discovered values for the zeta function at
In general for s an even integer, say
, for
, the zeta function
, evaluated at positive even integers takes on the values given by
(7)
where
are the Bernoulli numbers. These results were discovered by Leonhard Euler (1707-1783) sometime around 1724 and are well known. Observe that
is proportional to
.
The above results can be derived from the following observations. The function
can be expressed in different forms. For example,
Now compare the last term of the above equation with the previous equation (6) involving the Bernoulli numbers, to obtain
(8)
One can examine the zeros of the denominator in
and express
in the alternative form
The last term of the above equation can be expanded in a series to obtain
One can interchange the order of summation and write
(9)
Now by comparing the coefficients of powers
in the equations (9) and (8) one obtains the well known result
as previously given in equation (7). A similar derivation can be found in the reference [5].
4. The Zeta Function
Note that the reference [2] points out that there is no known formula for the zeta function evaluated at odd positive integers greater than or equal to three. This paper will provide such a formula.
It will be demonstrated that for odd positive integers s, say
, for
that
where the ellipsis
denotes the decimal representations are unending. In general, it will be demonstrated
(10)
where
are the Euler numbers and
are the polygamma functions. Observe the
is related to
. Apéry’s constant is
5. Polygamma Functions
The digamma function
is defined
(11)
where
is the gamma function.
(12)
The gamma function satisfies the functional equation
so one can write
(13)
and consequently
(14)
Take logarithms on both sides of equation (14) and then differentiate to show
Differentiate again and show
(15)
From the reference [2] or reference [4] (5.15), one can show that in the limit as n increases without bound the derivative term
behaves like 1/n and approaches zero. By repeated differentiation of equation (15) one can obtain the polygamma functions
defined by
(16)
for
.
6. Additional Functions
Related to the study of the zeta function are the Dirichlet1 eta, lambda and beta series defined
The first two Dirichlet series are related to the zeta function by the identities
(17)
Make note of the fact that knowing the equations (7) and (10) one can construct closed form expressions for the Dirichlet eta and lambda functions evaluated at odd and even integers greater than one.
7. Preliminary Observations
Define the function
(18)
where s is a positive integer greater than 1. One can then verify that
(19)
We examine the special cases
(20)
from which
can be obtained and
(21)
from which an expression for
, k an integer, can be obtained. From these two equations one can develop closed form expressions for
and
.
8. Calculation of
and
Observe that by using equation (16) with
, and again with
, one can obtain the series representations
(22)
These results will be used shortly.
9. Calculation of
and
We begin by examining the trigonometric function
which can be expressed in many different forms. One form is
where one can examine the zeros of the denominator and write
where
are constants which can be determined from the limits
This produces the expression
which can now be expanded into the series
(23)
Another form for
is
(24)
where the coefficients
are known as the Euler zigzag numbers. Still another form for
is
(25)
where
and
with
and
denoting the Euler and Bernoulli numbers.
Comparing like powers of x from equations (23) and (24) one can establish the relation
(26)
where the right-hand side of the equation is recognized as the
series or
series, depending upon the value of n. Replace n by 2n in equation (26) to obtain
(27)
Comparing like powers of x using the equations (24) and (25) one can show
(28)
which expresses the zigzag numbers in terms of the Euler and Bernoulli numbers. Therefore, the equation (27) can be expressed in the alternative form
(29)
a result also found in references [5] [6].
In equation (26) let
and show
and consequently the equation (21) can be written
(30)
giving a closed form expression for
where
. Note Catalan’s constant is given by
10. Calculation of
Use the results from equations (29), (20) and (22) one can demonstrate the equation
can be expressed in the form
for
. Solving for
one obtains the closed form expression given by equation (10) for the zeta function evaluated at odd positive integers greater than or equal to three.
11. Riemann Zeta Functional Equation
Several derivations of the Riemann zeta functional equation can be found in the reference [3]. One derivation is as follows. Using the definition of the gamma function
(31)
make the substitutions
and
to obtain after simplification
A summation of both sides of this equation over the index n and interchanging summation and integration one can show the above equation reduces to
(32)
Here
is the Riemann zeta function and
is related to the Jacobi theta function
(33)
Define
and express equation (33) in the form
The Jacobi theta function satisfies the property
which can be written in terms of the function
as
(34)
The equation (32) can now be expressed
The first integral on the right-hand side can be written in a different form as follows.
which simplifies to
This integral is further simplified by making the substitution
to obtain
This last integral allows one to express the equation (32) in the form
(35)
Observe that the right-hand side of equation (35) remains unchanged when s is replaced by
. This implies
(36)
which is the Riemann zeta functional equation. Multiplication of equation on both sides by
and using the Euler reflection formula
and the Legendre duplication formula
the functional equation can be expressed in the alternative form
which simplifies to
(37)
Replacing s by
the Riemann zeta functional equation can also be expressed in the form
(38)
12. Zeta Function for 0 and Negative Integers
The Riemann zeta functional equation is used to demonstrate
(39)
since
for all values of the integer n. These values for the zeta function are known as the trivial zeros. The nontrivial zeros lie in the complex plane. Also the Riemann zeta functional equation gives
Using the results from equation (7), this simplifies to
(40)
where
are the Bernoulli numbers.
Using the fact that
for odd integers greater than one the equations (39) and (40) can be combined into the form
(41)
for n a positive integer or zero.
This last equation also gives the integer values
(42)
Recall the value
does not exist as the series is the harmonic series which diverges for
. These values added to the values presented earlier will give the value of the zeta function at integer values, different from 1, along the real line.
For additional representations involving the zeta function in various forms and evaluated at other values the reader is referred to the references [2] [3] [6] [7] [8].
13. Zeros of the Zeta Function
The Euler product formula is used to demonstrate
whenever
. The Dirichlet eta function
is used to study the zeros of the zeta function for
,
, since it is related to the zeta function
(43)
The eta function is a converging alternating series for
and is sometimes referred to as the alternating zeta function. The equation (43) shows
whenever
. The factor (
) is zero at the points
, for all nonzero integer values for n. These are additional zeros of the eta function.
Writing
where for
one can show
(44)
and verify that
so the Cauchy-Riemann equations are
satisfied. This show
is an holomorphic function which satisfies
. This implies that if
for some value of s, then its conjugate
satisfies
. This demonstrates that the zeros of the zeta function are symmetric about the
-axis. The equation
is satisfied if both the real part u and imaginary part v of
are zero simultaneously. The condition
and
simultaneously is illustrated in Figure 1 by plotting
vs t in the special case where
. The special case
was selected for Figure 1 because of the Riemann hypothesis which is a conjecture that the nontrivial zeros of the zeta function have a real part equal to one-half. The values
are the values of t where
and
simultaneously for
. Here
is called the critical line and the region
is called the critical strip. To see the first one hundred imaginary parts of the complex zeros one can visit the web site https://wow.Imfdb.org/zeros/zeta/. A huge number of these complex zeros have been calculated and all lie on the critical line where
. Currently there is no proof that all of the nontrivial zeros of the zeta function must lie on the critical line.
Figure 1. Plot of
vs t, for
, with
.
14. Conclusion
A closed form expression for the Riemann zeta function evaluated at odd positive integers greater than three has been presented having the form
where
are the Euler numbers and
are the polygamma functions. It has been demonstrated that knowing closed form expressions for
and
for
one can construct closed form expressions for the Dirichlet eta, lambda and beta series at the even and odd integers different from unity. Closed form representations of the Apéry’s constant
and the Catalan’s constant
are obtained.
15. The Riemann Hypothesis
The Riemann hypothesis is a conjecture that all nontrivial zeros of the zeta function have a real part equal to one-half. If this is true, all nontrivial zeros are complex
numbers of the form
, called the critical line. Whether this is true or not is still an open question.
The Clay Mathematics Institute in Petersborough, New Hampshire is offering a one million dollar prize to anyone who can prove this conjecture and show how to calculate all the zeros of the zeta function. For additional information and conditions to be met in order to win the prize, the reader can consult reference [2] under the search name Riemann zeta function prize.
NOTES
1Peter Gustav Lejeune Dirichlet (1805-1859).