## across 3 years ago How many prime numbers are there between 1,000,000 and 1,100,000?

1. mathmate

Does that have to do with log(n)/n and the Riemann-Zeta function?

2. mathmate

Oops, n/log(n)

3. across

With this question, I am seeking to clash a mathematical approach with a computerized one. Although it would be interesting to see in how many different ways we can tackle this exercise.

4. mathmate

So that makes 6692.7, say 6693. Is there a better estimate using R-Z function, I don't really understand it.

5. across

mathmate, your approximation is fairly close!

6. across

For corroboration, I wrote this: http://ideone.com/qoDbe I am now trying an analytical approach.

7. mathmate

Can you elaborate on your analytical approach, or is it proprietary?

8. mathmate

6693 wasn't even close, you were being polite! Using the "offset logarithmic integral" $Li(n) = \int\limits_{2}^{n}\ \frac{dt}{\log(t)}$ I get 7212.99 > 7213. Using your code, I get 7216. So now it's getting close. Also, your code included 1 as a prime, which it is not. This does not change the counts over 1 though.

9. across

That made me want to go back and recheck the code. ^^

10. GT

wow.....

11. across

Interesting. Using the logarithmic integral from 2,000,000 to 2,100,000 you get 6881, whereas the (fixed) code chunks out 6871. The error is now negative.

12. GT

Code looks good except comparison to 1.

13. GT

btw - why picture change?

14. mathmate

I get the same answers, except that I get 6872 for the actual count, both using your code and my code.