Let's say you sit in a pub, minding your own business, when all of a sudden a stranger walks up to you and offers you a bet:
We'll choose two positive integers at random. If they have any divisor in common (other than ) I'll pay you a dollar, else you'll pay me a dollar. Are you in?
Apart from the question what kind of establishments you frequent, you should be wondering: is this a good bet for you?
When two integers have no divisors in common except the trivial divisor we say they are coprime or relatively prime. and have the common divisor , so they are not coprime, whilst and only have the trivial common divisor , so they are coprime.
This makes you start thinking: "As numbers grow bigger, aren't there a lot of divisors out there? After all, half the numbers are even, so if we hit two even numbers, they'll have the factor in common and I'll win. And then there's , , , ... Seems like a good deal!" Continue reading The Prime Bet
We've seen the calculus version
of the Euler product, and we know how to express as a product over its roots
High time we put everything together -- the reward will be the long expected explicit formula for counting primes! Continue reading Integral Madness
We know a lot about the and -functions, we've learnt all about the different prime counting functions, most notably , so it's high time we found a connection between the two. Probably not too surprisingly, the crucial link is our good friend, the Euler product
What we want to develop now is a version of this product that will suit us to find a formula that magically can count primes. (Remember that the Euler product is an analytical version of the fundamental theorem of arithmetic, so this is a natural starting point for our search.) Continue reading From Zeta to J and Back (And Yet Again Back)
Usually, I don't care too much about convergence as a general overview of the argument is what I aim at here, and otherwise I'll just trust that things "behave well". But some words concerning convergence won't harm.
It's a well known fact that the harmonic series (which we shortly touched in the previous post) diverges. I think the best (though not easiest) proof of this to compare it with the corresponding integral:
(Let's pause for a moment to celebrate the first of the numerous appearances of our good friend the logarithm.) Continue reading Does the Euler Product Converge?
From the previous post we know that the harmless looking series can be extended to the product . At first sight, this does not seem terribly helpful, and it actually makes the rather easy series more complicated. So what's the big deal?
It's what has actually been suppressed in the above notation: The sequences we run through. The series runs over all natural numbers (or positive integers, if you prefer), the product runs through all prime numbers. Now, that's cool, isn't it? We found a series over the natural numbers that, as we will see later, defines a well-behaved function which is accessible to all the nice methods modern mathematics can offer, and related it to the prime numbers.
In other words: Riemann's -function encodes the mysteries of the primes. Continue reading Euler Product Revisited
I will start this blog the way Bernhard Riemann started his paper: with Euler's product formula that John Derbyshire called the golden key:
This holds for any complex number with . If you look up a proof in any modern textbook, you will find a number technical rearrangements that end up in an examination of the absolute convergence on both sides. But actually, the formula is nothing but a fancy way of writing out the Sieve of Eratosthenes. Let's start by writing out the sum on the left hand side: Continue reading In the Beginning, There Was... Euler's Formula!