Category Archives: Riemann’s Paper

Applying the Explicit Formula

It's quite some time since we arrived at Riemann's main result, the explicit formula

J(x)=\mathrm{Li}(x)-\sum_{\Im\varrho>0}\left(\mathrm{Li}(x^\varrho)+\mathrm{Li}(x^{1-\varrho})\right)+\int_x^\infty\frac{\mathrm{d}t}{t(t^2-1)\log t}-\log2,

where J(x) is the prime power counting function introduced even earlier. It's high time we applied this!

First, let's take a look at J(x) when calculating it exactly:

J(x) Continue reading Applying the Explicit Formula

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




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

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