Gauss and the distribution of prime numbers
Teenage Carl Friedrich Gauss counted primes by hand and noticed that the probability any number near n is prime is approximately 1/ln(n). From that observation he built the prime counting function, estimating the number of primes below n as n/ln(n). He also noted that the logarithmic integral Li(n) tracks prime counts even more closely, though he wrongly assumed Li(n) always stays above the true count. These findings became the prime number theorem, setting the stage for Riemann's 1859 paper, which extended Euler's zeta function to complex variables.
- Gauss made these observations as a teenager, counting primes by hand, which means n/ln(n) was derived from pure empirical pattern-spotting rather than formal proof.
- The prime counting function is written pi(n), and n/ln(n) is described as asymptotic to it: the two curves stay proportionally close but n/ln(n) consistently undershoots.
- Li(n) overshoots the true prime count most of the time, but the hosts note it does fall below at numbers as large as 10 to the power of 10 to the 35, something Gauss could not have discovered by hand.
- Riemann's analytic number theory paper was the only one he ever wrote in that field; his other major work concerned geometry, including the Riemannian manifolds that later underpinned general relativity.
- The hosts mention the mathematical convention that results are named after the second discoverer because the first is almost always Euler, which is why the function is called the Riemann zeta function rather than Euler's.
Transcript
This chapter, from the episode video's captions · 680 words
22:08there's a teenage Carl Friedrich Gaus. >> Gaus. >> Gaus is another amazing mathematician. Um perhaps second or third only to Oiler and maybe Remon actually. Um he starts counting primes when he was a teenager cuz I guess that's what you do when you're a full autist in Germany. 17. >> Yeah. So, so he he starts counting the number of primes by hand. >> Unbelievable. >> And he notices that the probability that a certain number is going to be prime is related to the log of that number. It's basically one over the log. Like if I if I if I go up to like um a 10,000, right?
22:48>> The probability that numbers around 10 10,000 obviously is not prime, but like any given number around there is prime is 1 divided by the log of 10,000. log base e >> okay so from there he realizes that there's this prime counting function where I count how many primes before a certain number and that's related to n / log n okay there in the blue is you see a pi of n that's the prime counting function n / log n is in red and you can see that n over log n kind of undershoots he also realizes that the >> um the logarithmic integral meaning the integral of the log which is li of n that's kind of an overshoot most of the time. Turns out it's not all
23:28the time. If you go to 10 the 10 the 35 then prime numbers go above the the log integral. But like he he actually thought that perhaps the integral function is like always above. >> Mhm. >> But um he didn't know that 10 the 10 the 35 couldn't get that. >> Look [laughter] if you see if you do it by hand and you see a pretty good you're like it's probably you know I get it. So this becomes now the prime number theorem. Okay, which is that the number of primes before a certain level. The number of primes before a certain number is related to n / log n.
24:08Okay. >> Which is our red here. >> Yeah. Yeah. It's like asmtoic to n / like it has the same shape. Yes. >> Okay. It's not like going to diverge like crazy, right? It's always going to stay close in some sense. Right. Now enter Bernard Reman. >> Reman. [clears throat] >> Okay. >> Reman. >> Reman at the University of Berlin. He's got a 1859 article on the number of primes less than a given magnitude. Okay. This is the only um this is the only paper that he ever wrote in analytic number theory and it is probably his most influential. He was mostly like we're worried about geometry and things like that. Um like Remanian manifolds and stuff. That's the stuff that you get into with like general relativity. But this is what he wrote
24:50about um the prime number theorem and ever since then it has been named the reman zeta function function. Okay. Because there's a there's a joke in mathematics you always name something after the second person who discovered it because the first person is always Oiler and you can't name everything after Oiler you know. So some people call it the remon oiler zeta function but let's just give it to Right. Right. Right. Right. >> So >> we already know who the goat is. >> Yeah. Yeah. Exactly. >> We already know who the goat is. >> Exactly. So um he actually did some amazing work on the remon on the zeta
25:31function. What he did was extend Oilers's definition to the complex variables. What I mean by that is um you know usually we think about that zeta function as like one over the squares like one over um one over the squares or one over the cubes the the exponent is a real number you can even imagine one over square roots and things like that right um
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