Prime Number Theorem: the meaning of the limit

In summary, the conversation discusses the limit of the ratio between the prime counting function and the logarithmic integral function as n tends to infinity. It is mentioned that this limit does not imply that both functions converge to the same value, and that the growth rate of the two functions may be separated by a constant. It is also noted that the function for prime counting increases by one unit at a time and can be considered continuous for high values of n. However, it is also discussed that the growth rate of this function does not have a limit as n tends to infinity. The conversation also mentions the Riemann hypothesis and how it gives an upper limit on the difference between the two functions, but this limit still goes to infinity. Finally, the
  • #1
DaTario
990
33
Hi All.

I have a doubt concerning the limit:

$$ \lim_{n \to \infty} \frac{\pi (n)}{Li(n)} = 1 $$.

This mathematical statement does not imply that both functions converge to the same value. The main reason is that both tend to infinity as n tend to infinity. I would like to ask you if it is correct to infer that when n tends to infinity, ##\pi (n) ## and ## Li(n) ## grow at the same rate, possibly being separed by a constant.

If I am correct, is this constant known?
DaTario
 
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  • #2
How do you want to interpret the growth rate of ##\pi(n)##? It is a step function, which is discontinuous and hence constant except at a countable collection of points, at which it is not differentiable. But most interpretations of growth rate involve differentiation.
 
  • #3
DaTario said:
This mathematical statement does not imply that both functions converge to the same value. The main reason is that both tend to infinity as n tend to infinity. I would like to ask you if it is correct to infer that when n tends to infinity, ##\pi (n) ## and ## Li(n) ## grow at the same rate, possibly being separed by a constant.
The limit statement is even weaker. As an example: $$\lim_{n \to \infty} \frac{\ln (n) + \ln (\ln (n))}{\ln(n)} = 1$$
The difference between numerator and denominator diverges, but the ratio still goes to 1.

For this particular case, it is known that ##\pi(n) > Li(n)## and ##\pi(n) < Li(n)## switch infinitely often, the first time before 10371. Looks like there is no upper bound on the absolute deviation.
More about it at mathworld
 
  • #4
Thank you, mfb.

Answering andrewkirk, the function ##\pi (n)## is forced, by its definition, to increase by one unity at a time. As is goes to infinity, the relative weight of this discreteness goes to zero, allowing one to consider it as a continuous function. Its logarithmic nature also allows us to trust that no great departure from smoothness may happen for high values of ##n##. So the derivative of ##\pi(n)## may be taken as the average of:
$$\frac{\pi(n+1) - \pi(n)}{(n+1) - n} $$

for, say, a million of values of ##n## located at the vicinity of the point whose derivative we wish to calculate.
Note that our region of interest is ##n## large.

(someone, please correct me if I am wrong...)

Best wishes,

DaTario
 
  • #5
There is no need to smoothen the function arbitrarily.
 
  • #6
DaTario said:
the function ##\pi (n)## is forced, by its definition, to increase by one unity at a time. As is goes to infinity, the relative weight of this discreteness goes to zero, allowing one to consider it as a continuous function. Its logarithmic nature also allows us to trust that no great departure from smoothness may happen for high values of ##n##. So the derivative of ##\pi(n)## may be taken as the average of:
$$\frac{\pi(n+1) - \pi(n)}{(n+1) - n} $$

for, say, a million of values of ##n## located at the vicinity of the point whose derivative we wish to calculate.
Let's formalise that. Your definition of growth rate is then

$$G(n)=10^{-6}\times \sum_{k=1}^{1000000}\frac{\pi(n-k+1)-\pi(n-k)}{(n-k+1)-(n-k)}=
10^{-6}\times\Big(\pi(n)-\pi(n-1000000)\Big)$$

That will not attain any limit as ##n\to\infty## because it will be 0 for many large ##n## but ##10^{-6}## for some and, rarely, ##m\times 10^{-6}## for values of ##m## greater than 1.

The growth rate of ##Li(n)## is ##H(n)\equiv\frac d{dn}\int_2^n\frac1{\log t}dt= \frac1{\log n}##, which smoothly tends towards a limit of zero as ##n\to\infty##. So neither ##G(n)-H(n)## nor ##\frac{G(n)}{H(n)}## has a limit as ##n\to\infty##.
 
  • #7
Perhaps if we try to make a linear regression with a million of points of ##(n,\pi(n))##, with n ranging from ##n_0 - 5 \times 10^5 ## to ##n_0 + 5 \times 10^5 ##.
I would also accept a polynomial interpolation or a convolution with some gaussian.:smile:

My point is that I was taking the limit of the theorem as an inference that both function would be identical as n tends to infinity. I now see that what seems to happen is very different from this.
 
  • #8
DaTario said:
I now see that what seems to happen is very different from this.
Right. There are always points where the functions differ by more than 10, or more than 100, or probably even more than a million. Removing the steps of 1 doesn't help to get convergence.
 
  • #9
This consequence of the Riemann hypothesis gives some hint about this difference:

NumberedEquation20.gif
 
  • #10
It is an upper limit on the difference, but that limit goes to infinity. It is sufficient to see that the ratio converges to one, but it doesn't tell us if it has a fixed upper limit.
 
  • #11
Yes.

Thank you.
 
  • #12
Basically they try and use it to get the idea of how the prime counting function behaves.

The better the approximation - the better you can predict what the primes are.

The prime counting function though - is often elusive because of how "random" it seems.
 
  • #13
Thank you, chiro.
 

1. What is the Prime Number Theorem?

The Prime Number Theorem is a mathematical theorem that describes the distribution of prime numbers. It states that as the number of natural numbers increases, the proportion of those numbers that are prime approaches a limit of approximately 1/ln(n) where n is the number of natural numbers.

2. What is the significance of the limit in the Prime Number Theorem?

The limit in the Prime Number Theorem represents the proportion of prime numbers in relation to all natural numbers. It helps us understand the behavior of prime numbers and their distribution as we approach infinity.

3. How was the Prime Number Theorem discovered?

The Prime Number Theorem was first conjectured by mathematicians Adrien-Marie Legendre and Carl Friedrich Gauss in the late 18th and early 19th century. It was then proved by mathematician Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896.

4. What are some real-life applications of the Prime Number Theorem?

The Prime Number Theorem has been used to study and understand various areas of mathematics such as number theory, algebra, and analysis. It also has applications in cryptography, which is used in computer science and data security.

5. Are there any limitations to the Prime Number Theorem?

Yes, there are limitations to the Prime Number Theorem. It only gives a rough estimate of the distribution of prime numbers and does not provide an exact formula for finding prime numbers. It also does not predict the specific location of prime numbers, as they are randomly distributed among the natural numbers.

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