Zeta function

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 $1$) 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? [Read More]

Update 2024-10-20: Watch the video now in 4K! One stubborn source of frustration when working with complex numbers is the fact that visualisation becomes tedious, if not impossible. Complex numbers have 2 “real” dimensions in themselves, which give rise to the complex plane. That’s all good and fair. But if you talk about a function with complex domain and codomain, you already deal with a 4-dimensional graph. Unfortunately, my mind can only handle 3 dimensions (on a good day). [Read More]

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: You see how this jumps by one unit at prime values ($2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$), by half a unit at squares of primes ($4$, $9$), by a third at cubes ($8$), and by a quarter at fourth powers ($16$), but is constant otherwise. [Read More]