Integral Madness

We've seen the calculus version

J(x)=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\log\zeta(s)x^s\frac{\mathrm{d}s}{s},

of the Euler product, and we know how to express \xi(s) as a product over its roots

\xi(s)=\xi(0)\prod_\varrho\left(1-\frac{s}{\varrho}\right),

where

\begin{align}\xi(s)&=\frac{1}{2}\pi^{-s/2}s(s-1)\Pi(s/2-1)\zeta(s)\nonumber\\&=\pi^{-s/2}(s-1)\Pi(s/2)\zeta(s).\nonumber\end{align}

High time we put everything together -- the reward will be the long expected explicit formula for counting primes! Continue reading Integral Madness

From Zeta to J and Back (And Yet Again Back)

We know a lot about the \zeta and \xi-functions, we've learnt all about the different prime counting functions, most notably J(x), 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

\zeta(s)=\prod_{p}(1-p^{-s})^{-1}.

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)

Infinity Is Worth No More Than -1/12

On 16 January 1913, a confused manuscript reached the famous mathematician G. H. Hardy in Cambridge. Other researchers have received similar letters before, and rejected it due to the seemingly incoherent formulae mixed with trivial mathematical results. Professional mathematicians are used to receiving manuscripts by amateurs who believe to have solved famous problems, but this particularly odd scribble caught the eye:

1+2+3+4+5+6+\ldots+\infty=-\frac{1}{12}

Did this amateur mathematician really think that the sum of all natural numbers, a value that will exceed any given boundary at some point, will wind up being a negative fraction? M. J. M. Hill of the University College, London, simply responded that the author must have fallen victim to the pitfalls of divergent series and referred him to a standard textbook on the topic.

So, was this really just the work of a lunatic? Well, recently, the New York Times covered the topic, linking a video in which two physicists explain the importance of this result in modern string theory. While many physicists may not be too far from lunatics, these two make a pretty strong argument in this case: Continue reading Infinity Is Worth No More Than -1/12

Counting Primes Functionally

After all this playing with the \zeta-function it is time to return to the overall objective of this whole exercise: counting prime numbers. The idea behind analytic number theory is that primes are unpredictable on the small scale, but actually surprising regular on the large scale. This is why we'll look at certain functions that behave pretty erratically when we look at every single value, but become smooth and "easy" to calculate once we "zoom out" and consider the global properties, the so-called asymptotic. Continue reading Counting Primes Functionally

More symmetry and Another Product

We've seen that \zeta(s) satisfies the functional equation

\zeta(1-s)=2^{1-s}\pi^{-s}\cos(\pi s/2)\Pi(s-1)\zeta(s).

(Well, it still needs to be proved, but let's just assume it's correct for now.) The goal of this post is an even more symmetrical form that will yield the function \xi(s) which we can develop into an incredibly useful product expression.

On our wish list for \xi(s) we find three items:

  1. It's  an entire function, i.e., a function that's holomorphic everywhere in \mathbb{C} without any poles.
  2. It has zeros for all non-trivial zeros of the \zeta-function, and no others.
  3. It's perfectly symmetrical along the critical line, i.e., it satisfies \xi(1-s)=\xi(s).

Continue reading More symmetry and Another Product

Are Primes Independent?

The question may sound silly, but I hope it will become apparent that it's very reasonable to ask. What we will examine here is the probabilistic interpretation of the prime distribution. So, essentially we ask: "What's the probability that a randomly chosen number is prime?" Those familiar with some basic probability theory know the notion of independency in this context, so the question I'm basically interested in here is if the probability to find a prime is independent of the preceding or following numbers. This post is a little out of the regular narrative, but it's something I have been thinking about recently, and I think it makes a cute article.

But one step back. First consider the following problem: We pick a number X not exceeding some (large) n at random. What are the odds this number is prime? Well, this is a classic example of a discrete probability, so the answer is positive cases over all cases, i.e.,

P(X\in\mathbb{P})=\frac{\pi(n)}{n},

where \mathbb{P} denotes the set of all primes, and \pi(n) is the prime counting function, i.e., the number of primes not exceeding n. Now, the famous Prime Number Theorem states that we can approximate \pi(n) with the elementary function

\pi(n)\sim\frac{n}{\log n}.

Don't worry too much about the exact meaning of the tilde, we will have plenty of opportunity to examine it further. For now just read it as "the left-hand side behaves similarly to the right-hand side". Now we can substitute this approximation in the calculation of the probability above, and obtain

P(X\in\mathbb{P})\sim\frac{n/\log n}{n}=\frac{1}{\log n}.

Continue reading Are Primes Independent?

Perfect Symmetry

So far, we have seen how the Euler product links the \zeta-function to the prime numbers. More precisely, it encodes the fundamental theorem of arithmetic. One may also say, it's the analytic version of it, in a sense that should become clearer shortly.

What we have done so far works perfectly for the real numbers. The sum \sum n^{-s} that defines \zeta(s) converges for s>1, that's how Leonhard Euler found his product, and that's what Peter Gustav Lejeune Dirichlet used to prove the prime number theorem in arithmetic progressions. The ingenious step In Riemann's 1859 paper was to allow for complex values s. As mentioned before, the same argument as in the real case proves that the sum converges for \Re s>1. Since the convergence is absolute, \zeta(s) is analytic in this domain. If you don't know what analytic is, just read it as well-behaved or, even better, cool.

The shame is that -- as you certainly have already heard if you still read this blog -- all the action takes place in the critical strip, i.e. the area just to the left of our line of convergence with 0\le\Re s\le1. Riemann showed that we can calculate values of \zeta(s) for \Re s\le1 through the beautiful functional equation

\zeta(1-s)=2^{1-s}\pi^{-s}\cos(\pi s/2)\Pi(s-1)\zeta(s).

Continue reading Perfect Symmetry

Does the Euler Product Converge?

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) \sum1/n diverges. I think the best (though not easiest) proof of this to compare it with the corresponding integral:

\sum_{n=1}^x\frac{1}{n}>\int_1^x\frac{1}{t}\mathrm{d}t=\log x\longrightarrow\infty.

(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?

Euler Product Revisited

From the previous post we know that the harmless looking series \sum n^{-s} can be extended to the product \prod (1-p^{-s})^{-1}. 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 \zeta-function encodes the mysteries of the primes. Continue reading Euler Product Revisited

In the Beginning, There Was... Euler's Formula!

I will start this blog the way Bernhard Riemann started his paper: with Euler's product formula that John Derbyshire called the golden key:

\zeta(s)=\sum_{n\ge1}n^{-s}=\prod_{p}(1-p^{-s})^{-1}

This holds for any complex number s with \Re s > 1. 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!