The Riemann Zeta Function is defined by the following series:

Here s is a complex number and the first obvious issue is to find the domain of this function, that is, the values of s where the function is actually defined. First of all, it is a well known result in calculus that, when s is real, the series is convergent for s>1 (see [2]). For example, a simple application of the theory of Fourier series allows to prove that . For s=1, the series diverges. However, one can prove that the divergence is not too bad, in the sense that:

In fact, we have the inequalities:

Summing from 1 to , we find that and so

which implies our claim.

As a function of the real variable s, is decreasing, as illustrated below.

for s real and >1

The situation is more complicated when we consider the series as a function of a complex variable.

Remember that a complex number is a sum , where are real numbers (the real and the imaginary part of z, respectively) and , by definition. One usually writes There is no ordering on the complex numbers, so the above arguments do not make sense in this setting. We remind that the complex power is defined by

and

Therefore, the power coincides with the usual function when s is real.

It is not difficult to prove that the complex series is convergent if Re(s)>1. In fact, it is absolutely convergent because

where |z| denotes as usual, the absolute value: . See [2] for the general criteria for convergence of series of functions.

Instead, it is a non-trivial task to prove that the Riemann Zeta Function can be extended far beyond on the complex plane:

Theorem. There exists a (unique) meromorphic function on the complex plane, that coincides with , when Re(s)>1. We will denote this function again by

We have to explain what meromorphic means. This means that the function is defined, and holomorphic (i.e. it is differentiable as a complex function), on the complex plane, except for a countable set of isolated points, where the function has a pole. A complex function f(z) has a pole in w if the limit exists and is finite for some integer m. For example, has a pole in s=1.

It is particularly interested to evaluate the Zeta Function at negative integers. One can prove the following: if k is a positive integer then

where the Bernoulli numbers are defined inductively by:

Note that : the Bernoulli numbers with odd index greater than 1 are equal to zero. Moreover, the Bernoulli numbers are all rational.

Of course, the number is not obtained by replacing s=1-k in our original definition of the function, because the series would diverge; in fact, it would be more appropriate to write where the superscript * denotes the meromorphic function whose values are defined, only when Re(s)>1, by the series .

There is a corresponding formula for the positive integers:

2

It is a remarkable fact that the values of the Riemann Zeta Function at negative integers are rational. Moreover, we have seen that if n>0 is even. The natural question arises: are there any other zeros of the Riemann Zeta Function?

Riemann Hypothesis. Every zero of the Riemann Zeta Function must be either a negative even integer or a complex number of real part = ½.

It is hard to motivate this conjecture in an elementary setting, however the key point is that there exists a functional equation relating and (in fact, such a functional equation is exactly what is needed to extend to the complex plane). The point is the center of symmetry of the map

It is also known that has infinitely many zeros on the critical line Re(s)=1.

Why is the Riemann Zeta function so important in mathematics? One reason is the strict connection with the distribution of prime numbers. For example, we have a celebrated product expansion:

where the infinite product is extended to all the prime numbers and Re(s)>1. So, in some sense, the Riemann Zeta function is an analytically defined object, encoding virtually all the information about the prime numbers. For example, the fact that can be used to prove Dirichlets theorem on the existence of infinitely many prime numbers in arithmetic progression.

The product expansion implies that for every s such that Re(s)>1. In fact, we have:

and it is not difficult to check that this product cannot vanish.

The following beautiful picture comes from Wikipedia.

Bibliography

[1] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 2000

[2] W. Rudin, Principles of Mathematical Analysis, McGraw Hill, 1976

[3] W. Rudin, Real and Complex Analysis, , McGraw Hill, 1986

Here s is a complex number and the first obvious issue is to find the domain of this function, that is, the values of s where the function is actually defined. First of all, it is a well known result in calculus that, when s is real, the series is convergent for s>1 (see [2]). For example, a simple application of the theory of Fourier series allows to prove that . For s=1, the series diverges. However, one can prove that the divergence is not too bad, in the sense that:

In fact, we have the inequalities:

Summing from 1 to , we find that and so

which implies our claim.

As a function of the real variable s, is decreasing, as illustrated below.

for s real and >1

The situation is more complicated when we consider the series as a function of a complex variable.

Remember that a complex number is a sum , where are real numbers (the real and the imaginary part of z, respectively) and , by definition. One usually writes There is no ordering on the complex numbers, so the above arguments do not make sense in this setting. We remind that the complex power is defined by

and

Therefore, the power coincides with the usual function when s is real.

It is not difficult to prove that the complex series is convergent if Re(s)>1. In fact, it is absolutely convergent because

where |z| denotes as usual, the absolute value: . See [2] for the general criteria for convergence of series of functions.

Instead, it is a non-trivial task to prove that the Riemann Zeta Function can be extended far beyond on the complex plane:

Theorem. There exists a (unique) meromorphic function on the complex plane, that coincides with , when Re(s)>1. We will denote this function again by

We have to explain what meromorphic means. This means that the function is defined, and holomorphic (i.e. it is differentiable as a complex function), on the complex plane, except for a countable set of isolated points, where the function has a pole. A complex function f(z) has a pole in w if the limit exists and is finite for some integer m. For example, has a pole in s=1.

It is particularly interested to evaluate the Zeta Function at negative integers. One can prove the following: if k is a positive integer then

where the Bernoulli numbers are defined inductively by:

Note that : the Bernoulli numbers with odd index greater than 1 are equal to zero. Moreover, the Bernoulli numbers are all rational.

Of course, the number is not obtained by replacing s=1-k in our original definition of the function, because the series would diverge; in fact, it would be more appropriate to write where the superscript * denotes the meromorphic function whose values are defined, only when Re(s)>1, by the series .

There is a corresponding formula for the positive integers:

2

It is a remarkable fact that the values of the Riemann Zeta Function at negative integers are rational. Moreover, we have seen that if n>0 is even. The natural question arises: are there any other zeros of the Riemann Zeta Function?

Riemann Hypothesis. Every zero of the Riemann Zeta Function must be either a negative even integer or a complex number of real part = ½.

It is hard to motivate this conjecture in an elementary setting, however the key point is that there exists a functional equation relating and (in fact, such a functional equation is exactly what is needed to extend to the complex plane). The point is the center of symmetry of the map

It is also known that has infinitely many zeros on the critical line Re(s)=1.

Why is the Riemann Zeta function so important in mathematics? One reason is the strict connection with the distribution of prime numbers. For example, we have a celebrated product expansion:

where the infinite product is extended to all the prime numbers and Re(s)>1. So, in some sense, the Riemann Zeta function is an analytically defined object, encoding virtually all the information about the prime numbers. For example, the fact that can be used to prove Dirichlets theorem on the existence of infinitely many prime numbers in arithmetic progression.

The product expansion implies that for every s such that Re(s)>1. In fact, we have:

and it is not difficult to check that this product cannot vanish.

The following beautiful picture comes from Wikipedia.

Bibliography

[1] K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 2000

[2] W. Rudin, Principles of Mathematical Analysis, McGraw Hill, 1976

[3] W. Rudin, Real and Complex Analysis, , McGraw Hill, 1986