Prime Number Distribution

Question

Lists of prime numbers are often given 'up to' some number. (x primes up to 100,000, y primes up 1,000,000 etc.)

Yet their distribution is connected to processes that are frequency-based and thus logarithmic in the same way as musical notes. (Thus there are twice as many frequencies between C2 and C3 as between C1 and C2 etc)

Do you know of any analysis that has been done on the quantity of primes within successive 'octaves' of the number line? For instance, this might be an analysis of the increase in the size of the segment of the number-line containing 25 primes. For the first 25 primes it would be 100, then the next 25 would take us from 100 to 200+, and so on.

I suppose one could call this a dynamic analysis. I haven't seem such a thing but presume I just haven't looked in the right place.

Apologies if the question is not clear. I'm not a mathematician. I'll edit as advised.

Answer 1

Well, the concept of octaves makes me think of modular arithmetic. For instance, the interval of a tenth is equivalent to the interval of a third, and we see that $10 \equiv 3 \pmod 7$ (there are seven diatonic notes in the octave).

Since $\phi(100) = 40$, this suggests that in each interval of $100$ there may be as many as $40$ primes. The first interval, from $1$ to $100$ is unique because it includes two primes that are factors of $100$, and so, for example, $102$ and $105$ are not prime.

Answer 2

The other commenters are right that you can only answer these sorts of questions using the PNT but how to carry that out then is not immediately clear. Luckily Wikipedia has a very short section on this: https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number

In particular they state that

$$p_n \sim n \log n$$

where $p_n$ is the $n$'th prime number and $\log$ is the natural logarithm, written '$\log$' by mathematician and '$\ln$' by people in a bunch of other professions.

This approximation is not perfect, for instance it tells you that to get the first 25 prime numbers you should look between $0$ and $25 \log 25 \approx 80.5$, to get the next 25 you should look between $80.5$ and $50 \log(50) \approx 195.6$, to get the next 25 we should look between $195.6$ and $323.8$, etc.

These are all underestimations but they do nicely confirm your intuition that the intervals get longer and longer. On the other hand, whereas the distance you have to travel on the frequency scale in music to 'catch' 12 tones doubles each time, the distance you have to travel on the number scale (in math) to catch 25 primes does not double at all. Yes it increases but much much more slowly than the exponential growth we see in music. (You can see that of course in tables of primes, but the imperfect but simple formula above has the advantage that you can quickly compute its behavior up to very large numbers using a calculator or computer and check the rate of growth there.)

So, in short, this is not what the phrase 'music of the primes' refers to.

On the most superficial level we can say that what people mean when they draw an analogy between music and the distribution of primes is

"both can be understood better by using Fourier analysis".

Now I am not sure if there really is a deeper level on which the analogy holds, or that the statement in yellow is really all there is to it. But the best way to judge that for yourself is to understand how Fourier analysis is used in music and how it used in number theory. You probably already know the answer to the first of these questions and your series of questions here on MSE is slowly closing in on the second.