## What is pi(x)?

Limits, differentiation, related rates, integration, trig integrals, etc.
leelguy
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### What is pi(x)?

I was looking into prime numbers and i keep coming upon pi(x). Specifically http://primes.utm.edu/howmany.shtml. It is obvious that they arent multiplying pi*x. So what are they doing?

Thanks!

stapel_eliz
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leelguy wrote:I was looking into prime numbers and i keep coming upon pi(x). Specifically http://primes.utm.edu/howmany.shtml. It is obvious that they arent multiplying pi*x. So what are they doing?

I'm not sure what you mean by what they are "doing"...?

The referenced article mentions the name, "pi", of a function of a variable, "x"; they've used function notation to express the relationship between a number "x" and the number of primes "p(x)" that are less than or equal to "x". This formula is known as the Prime Counting Function.

Does that help?

leelguy
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### Re: What is pi(x)?

I have looked at the prime counting function, however it doesnt seem to work. For instance if I wanted to find how many primes are less than 45,000 then I tried 45,000/ln(45,000) and that didnt come up with the right answer. I dont really understand the wiki article on the prime counting function. If you could explain how to find the number of primes below a number in terms that someone in trig could understand then that would be awesome.

Thanks!

stapel_eliz
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leelguy wrote:I have looked at the prime counting function, however it doesnt seem to work. For instance if I wanted to find how many primes are less than 45,000 then I tried 45,000/ln(45,000) and that didnt come up with the right answer.

Part of the problem with counting primes is precisely that there is no formula that gives the answer. The function $\pi (x)$ is the actual count of the primes, but to find the value of $\pi (n)$ for any given $n$, you actually have to do the counting. This obviously can quickly become unreasonably time-consuming.

The other functions:

. . . . .$\frac{x}{\ln{(x)}}\, \mbox{ and }\, \int_0^x \frac{dt}{\ln {(t)}}$

...are approximations to $\pi (x)$. They are useful in that they give approximate values in the form of general formulas, rather than in the form of a laborious step-by-step counting.

leelguy wrote:If you could explain how to find the number of primes below a number in terms that someone in trig could understand then that would be awesome.

I'm sorry, but it is not reasonably feasible to attempt to teach classes within this environment.

But after you take calculus, some of the discussion of integrals, limits, etc, may make more sense.

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