Read Prime Obsession over the last five days (I got a hold of it just in time for vacation). It is a really nicely written book on the history and math behind the Riemann hypothesis. The areas of math involved here are number theory (prime numbers), and complex analysis. The Euler product (shown on the cover) is the beginning of the bridge between the two fields.

Mathematicians are interested in the number of primes less than a given integer and call this . While the function can be calculated for any given , there is no closed form (simple) solution for it. Approximations have been proposed, the first being

and the second being

and the closest approximation coming in the last chapter.

Meanwhile, also of interest are the convergent infinite series of fractions, given by the Riemann zeta function . This function is defined as

The first solution to this I encountered in college for which is amazingly

Euler solved this generally for all even, (and they are all powers of to the ). However, all odd are unsolved, and it was not until 1978 that was even proved to be irrational! This summed series can be converted with regular algebra (and some insight using something like the sieve of Eratosthenes) to a product series. This is the famous **Euler product**

where are all the integers and stands for all prime numbers. (The use of is a convention started by Riemann, and not disturbed by mathematicians over the centuries, it could just as well be ). The Riemann zeta function is continued analytically into the complex plane and gives rise to the Riemann hypothesis,* *

*The Riemann hypothesis – all non-trivial zeros of the zeta function have real part one-half.*

(now the zeta function in the complex plane is linked with primes!)

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