Riemann Hypothesis

The Riemann hypothesis, proposed by Bernhard Riemann in 1859, is a famous conjecture that has far-reaching consequences in many areas of mathematics. A proof of the Riemann hypothesis is one of the seven Millennium Prize Problems in mathematics. The first person to provide a proof will be awarded a prize of U.S. $1 million by the Clay Mathematics Institute.

Prime Counting Function

Definition

For an integer , the number of primes in the interval is denoted by the prime counting function

The following table gives the values of for

	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
	1	2	2	3	3	4	4	4	4	5	5	6	6	6	6	6	7	8	8

Mathematicians are interested in developing formulas for .

Notation

We will denote the -th prime number by , for integers . For example, , etc

Background

Lemma 1

For all integers :

Proof

Proof by induction on :
Let be the open sentence:

(see Summation and Product Notation)

Base Case:

Inductive Step:
Let
Assume the inductive hypothesis
We wish to prove

which is exactly the right hand side of the equation

The result is true for and so holds for all by POMI.

Upper Bound

Lemma

For all positive integers , we have

Proof

Proof by strong induction on , where is the statement

Base Case:
The statement is . Since , is true.

Inductive Step:
Let be an arbitrary natural number.
Assume the inductive hypothesis,
We wish to prove

Now, by Euclid's theorem, we know that the integer is divisible by a prime that is greater than . Hence,

The result is true for and holds true for all by POSI.

Lower Bound

Theorem

For all integers , we have

(see Logarithms)

Proof

Let be an arbitrary integer such that , and let be the integer satisfying:

We note that . By the upper bound lemma, we have , and hence .
Now, taking logarithms of bth sides of the inequality yields .
Since , we have , and taking logarithms again yields and therefore .
It follows that .

Better Lower Bound

Theorem

For all integers , we have .

Proof

Let be an arbitrary integer such that .
Define so gives all the prime numbers that are less than or equal to .
Now, for each integer , we can uniquely write:

where and are positive integers, and is squarefree (i.e not divisible by the square of a prime).
Thus, we have

and so .
Since , every prime factor of must be at most , and therefore at most . Thus, since is squarefree, we can write

where each .

Now, there are at most positive integers satisfying .
Also, there are at most positive integers satisfying (1).
Hence, there are at most integers of the form where are positive integers and is squarefree.
Since there are exactly integers in , we must have

and therefore . Taking logarithms of both sides gives , and hence we obtain

Prime Number Theorem

Theorem

(see Limits)

Remark

The prime number theorem describes the asymptotic distribution of prime numbers.
That is, it gives a formula for that increases in accuracy as gets larger.

Riemann Hypothesis

Conjecture

Let for . For all integers , we have

(see definite integrals)

Let us compare the estimates for from each theorem and the Riemann Hypothesis. We will consider the number of primes that are . The exact value for , obtained by a computer search, is

First, with the weak lower bound theorem:

And then with the better lower bound theorem:

With Prime number theorem:

which is much closer, agreeing on the first 2 digits.

Finally, with the Riemann Hypothesis:

Which is a very close estimate, agreeing on the first 14 digits.