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  . One way of defining this function is to express it as an infinite series having the form
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
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
where is the gamma function. Several proofs of the above result can be found in the Titchmarsh reference . 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
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
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  (24.2), where one finds the generating functions
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
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
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
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 .
4. The Zeta Function
Note that the reference  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
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
where is the gamma function.
The gamma function satisfies the functional equation so one can write
Take logarithms on both sides of equation (14) and then differentiate to show
Differentiate again and show
From the reference  or reference  (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
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
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
where s is a positive integer greater than 1. One can then verify that
We examine the special cases
from which can be obtained and
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
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
Another form for is
where the coefficients are known as the Euler zigzag numbers. Still another form for is
with and denoting the Euler and Bernoulli numbers.
Comparing like powers of x from equations (23) and (24) one can establish the relation
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
Comparing like powers of x using the equations (24) and (25) one can show
which expresses the zigzag numbers in terms of the Euler and Bernoulli numbers. Therefore, the equation (27) can be expressed in the alternative form
a result also found in references  .
In equation (26) let and show
and consequently the equation (21) can be written
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 . One derivation is as follows. Using the definition of the gamma function
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
Here is the Riemann zeta function and is related to the Jacobi theta function
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
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
Observe that the right-hand side of equation (35) remains unchanged when s is replaced by . This implies
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
Replacing s by the Riemann zeta functional equation can also be expressed in the form
12. Zeta Function for 0 and Negative Integers
The Riemann zeta functional equation is used to demonstrate
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
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
for n a positive integer or zero.
This last equation also gives the integer values
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     .
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
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
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 .
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  under the search name Riemann zeta function prize.
1Peter Gustav Lejeune Dirichlet (1805-1859).