Stumbling through mathematics: the Riemann hypothesis

The Riemann zeta function ζ(s), where s is a complex variable, is given by:

Leonhard Euler (1707-1783) discovered that ζ(s) could be represented as an infinite product involving the prime numbers:

where p represents the prime numbers 2, 3, 5, ... The "trivial" zeros of the zeta function are ζ(-2) = 0, ζ(-4) = 0, ζ(-6) = 0, etc. The following is a Maple plot of a few of the trivial zeros:

The Riemann hypothesis states that all of the non-trivial zeros of the Riemann zeta function line on the line Re(s) = 1/2. Below is a plot of |ζ(1/2 + iy)| which shows a few of the non-trivial zeros:

As of 2004 over 10,000,000,000,000 non-trivial zeros have been calculated. All of them lie on the line Re(s) = 1/2; however, this does not constitute proof of the Riemann hypothesis. To date no one has been able to prove (or disprove) the Riemann hypothesis. A $1M prize has been offered to anyone who offers a valid proof one way or the other.

An excellent account of the Riemann hypothesis and its history is Prime Obsession, by John Derbyshire.

Stumbling through mathematics: strange powers

You may have been told by your teacher that square roots of negative numbers do not exist. But what if they actually did? It turns out that the square roots of negative numbers are surprisingly useful in mathematics and engineering.

High school and college students may remember this strange creature from their math classes: √(-1). If you try putting this number into your calculator, you will likely get an error message. So what is it? Mathematicians call this number an imaginary number, and denote it using the symbol i = √(-1).

It turns out that this mysterious number i has some interesting (as well as useful) properties. For example: what happens when we raise i to the second power, the third, etc.?

i¹ = i
i² = (√(-1))² = i² = -1
i³ = (√(-1))²⋅√(-1) = i²⋅i = -1⋅i = -i
i⁴ = (√(-1))³⋅√(-1) = -i⋅i = -i² = -(-1) = 1
i⁵ = (√(-1))⁴⋅√(-1) = 1⋅i = i
i⁶ = (√(-1))⁵⋅√(-1) = i⋅i = i² = -1
i⁷ = (√(-1))⁶⋅√(-1) = i⁶⋅i = -1⋅i = -i
i⁸ = (√(-1))⁷⋅√(-1) = -i⋅i = -i² = -(-1) = 1


...and so on. Notice how the pattern of i, -1, -i, 1 keeps repeating itself? This repetition is one property that makes imaginary numbers so useful.

But what happens if we raise i to an imaginary power, say, i ⁱ ? What happens then?

To see what i ⁱ is, we must use Euler's equation (an equation familiar to physicists and electrical engineers as well as mathematicians). Let e be the base of the natural logarithm (approximately equal to 2.7182....), and cos(x) and sin(x) be the cosine and sine functions of x:

e^ix = cos(x) + isin(x)

A complex number is a number made up of a real number and an imaginary number. Complex numbers have the form a + bi or a - bi, where a and b are real numbers. Any real number can be written as a complex number. An example would be the real number 3, which can be written in two different (but equally valid) ways: 3 + 0i or 3 - 0i. All your life you have been using complex numbers - and perhaps did not even know it!

Therefore, i can be written as the complex number 0 + 1⋅i. One value of x that would make cos(x) = 0 and sin(x) = 1 is x = π/2. Then Euler's equation becomes:

e^iπ/2 = 0 + 1i = i

Now raise both sides on this equation to the i power:

(e^iπ/2)ⁱ = e^i2π/2 = i ⁱ

Since i ² = -1:

e^-π/2 = i ⁱ

On the right side of this equation we have an imaginary number being raised to an imaginary power. On the left side we have a real number with no imaginary part whatsoever (the value of e^-π/2 is an irrational number and is approximately equal to 0.2078795764...). This result is counter intuitive, to say the least!

But that is not the end of the story. The value x = π/2 is only one possible result. Because cos(x) and sin(x) are periodic functions, there are many values of x that make cos(x) = 0 and sin(x) = 1. In fact, there are an infinite number of values of x! We can re-write the cosine and sine terms to reflect this: cos(π/2 ± 2πn) = 0 and sin(π/2 ± 2πn) = 1, where n is any non-negative integer such as 0, 1, 2, 3, .... The equation can now be re-written as:

e^{-π/2 ±2πn} = i ⁱ

e^-π/2⋅e^±2πn = i ⁱ

So, i ⁱ results in an infinite number of real solutions. There are many strange results in mathematics, but I would be hard-pressed to find one stranger than this. It is why mathematics is the best science ever!

Christians in mathematics: Leonhard Euler

Leonhard Euler may have been the most famous mathematician of all times. He was certainly one of the most prolific writers. For nearly fifty years after his death the St. Petersburg Academy (where he had been the head of the mathematics department) continued to publish his unreleased works.

As a child Euler had shown an unusual amount of talent in the field of mathematics, a fact that was all the more remarkable since he had mostly studied the subject on his own or with private tutors. Euler's father, a minister, had wanted his son to enter the ministry and sent him to the University of Basel in Switzerland where in 1723 he completed his Master's degree in philosophy . However, Johann Bernoulli (another famous mathematician) eventually convinced the elder Euler to let his son study mathematics instead of theology.

Although he never became a minister Euler was a devout Christian all of his life. Once, while he was working in the court of Catherine the Great, Euler had a debate with the atheist French philosopher Denis Diderot over the existence of God. Euler claimed that he had a mathematical proof which he stated as follows:

Diderot, whose knowledge of mathematics was minimal, soon left St. Petersburg and returned to France.

Euler and his wife had 13 childern, although only five lived past infancy. Euler was devoted to his family, and later said that he made mathematical discoveries while holding an infant in one arm.

Euler's talents were not confined to mathematics. He also made contributions to the fields of physics, astronomy, cartography and music.