The Golden Key to Riemann’s Hypothesis

The Riemann Hypothesis

Perhaps the greatest unsolved problem in mathematics is to explain the distribution of the prime numbers. The overall “thinning out” of the primes less than some number ${N}$ , as ${N}$ increases, is well understood, and is demonstrated by the Prime Number Theorem (PNT). In its simplest form, PNT states that the number of primes less than ${N}$ is approximately equal to ${N/\log N}$ . This implies that, for ${N}$ large, ${p_N \sim N\log N}$ and the probability that a number randomly chosen in the range from ${1}$ to ${N}$ is prime is approximately ${1/\log N}$ .

But hold on: don’t people say that the greatest unsolved problem is to prove the Riemann Hypothesis. How can both of these problems be “the greatest”? In fact, the distribution of prime numbers is intimately connected with the distribution of zeros of Riemann’s zeta function.

One of Euler’s early triumphs was to find the sum of reciprocals of squares of the natural numbers, ${\sum(1/n^2) = \pi^2/6}$ . Later, Euler studied the infinite series

$\displaystyle \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$

for real value of ${s}$ greater than ${1}$ . But the huge potential of this series becomes apparent only when ${s}$ varies freely in the complex plane.

Riemann first considered this generalization, and the function ${\zeta(s)}$ is now called the Riemann zeta function. It is relatively easy to show that ${\zeta(s)=0}$ for negative even numbers ${s=-2n}$ . These are known as the trivial zeros. All the remaining zeros are in the critical strip ${0<\Re\{s\} < 1}$ , and the Riemann Hypothesis is that all these non-trivial zeros lie on the critical line ${Re\{s\}} = \frac{1}{2}$ .

The connection between the Riemann Hypothesis and the distribution of primes is a fascinating and intricate story, beautifully described in an excellent book, Prime Obsession, by John Derbyshire (2003). To convince you that the two problems are intertwined, we will examine the wonderful product formula first proved by Euler.

Euler’s Product Formula: the Golden Key

Euler’s *“Series Infinitas”*, St Petersburg, 1737.

We start with the definition of the zeta function:

$\displaystyle \zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \frac{1}{4^s} + \frac{1}{5^s} + \frac{1}{6^s} + \frac{1}{7^s} + \frac{1}{8^s} + \cdots \,.$

Although ${s}$ may be complex, it is sufficient here to regard it as a real variable. If we multiply the series by ${1/2^s}$ , we get

$\displaystyle \frac{1}{2^s}\zeta(s) = \frac{1}{2^s} + \frac{1}{4^s} + \frac{1}{6^s} + \frac{1}{8^s} + \frac{1}{10^s} + \frac{1}{12^s} + \frac{1}{14^s} + \frac{1}{16^s} + \cdots$

Now, subtracting this from the expression for ${\zeta(s)}$ , we get

$\displaystyle \left( 1 - \frac{1}{2^s} \right) \zeta(s) = 1 + \frac{1}{3^s} + \frac{1}{5^s} + \frac{1}{7^s} + \frac{1}{9^s} + \frac{1}{11^s} + \frac{1}{13^s} + \frac{1}{15^s} + \cdots$

All the terms whose denominators have powers of even numbers are gone! Now multiply this by ${1/3^s}$ and subtract as before. We get

$\displaystyle \left( 1 - \frac{1}{3^s} \right) \left( 1 - \frac{1}{2^s} \right) \zeta(s) = 1 + \frac{1}{5^s} + \frac{1}{7^s} + \frac{1}{11^s} + \frac{1}{13^s} + \frac{1}{17^s} + \frac{1}{19^s} + \frac{1}{23^s} + \cdots$

All the multiples of ${3}$ have now vanished from the right side. Now multiply this by ${1/5^s}$ and subtract as before. We get

$\displaystyle \left( 1 - \frac{1}{5^s} \right) \left( 1 - \frac{1}{3^s} \right) \left( 1 - \frac{1}{2^s} \right) \zeta(s) = 1 + \frac{1}{7^s} + \frac{1}{11^s} + \frac{1}{13^s} + \frac{1}{17^s} + \frac{1}{19^s} + \frac{1}{23^s} + \frac{1}{29^s} + \cdots$

Continuing indefinitely, all the terms on the right after ${1}$ are obliterated so that, ultimately, we have

$\displaystyle \cdots\ \left( 1 - \frac{1}{5^s} \right) \left( 1 - \frac{1}{3^s} \right) \left( 1 - \frac{1}{2^s} \right) \zeta(s) = 1$

and rearranging this gives an expression for ${\zeta(s)}$ :

$\displaystyle \zeta(s) = \left( \frac{2^s}{ 2^s - 1} \right) \left( \frac{3^s}{ 3^s - 1} \right) \left( \frac{5^s}{ 5^s - 1} \right) \cdots\$

This may be written as an equation relating an infinite sum over all natural numbers and an infinite product over all prime numbers:

$\displaystyle \sum_{n=1}^\infty \frac{1}{n^s} = \prod_{p\in\mathbb{P}} \ \frac{p^s}{ p^s - 1} \,.$

This formula, which Derbyshire calls Euler’s Golden Key, shows that there must be an intimate connection between the zeta function (on the left) and the set of prime numbers (on the right).

All that remains is to complete the proof of the hypothesis but, given that it has defied the most brilliant mathematicians for one and a half centuries, we may have to wait some time.

Sources

${\bullet}$ Derbyshire, John, 2003: Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. Plume Books, 422pp. ISBN: 978-0-4522-8525-5.

${\bullet}$ Euler Leonhard, 1737. Variae observationes circa series infinitas (Various Observations about Infinite Series), Royal Imperial Academy, St Petersburg.

${\bullet}$ Riemann, Bernhard (1859): On the Number of Prime Numbers less than a Given Quantity. In Gesammelte Werke, Teubner, Leipzig (1892), Original manuscript and English translation here.

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