We know since ancient greeks that:
$$\pi(N)=N - \sum_p \left\lfloor \frac N p \right\rfloor + E(N)$$
Who was the first one to bound that $E(N)$? Which advances on it were made since then?
About an error term in Prime Counting Function
2
$\begingroup$
prime-numbers
math-history
distribution-of-primes
-
2$\sum_p \lfloor \frac {N} {p} \rfloor $ isn't a useful approximation to the prime counting function. I think already Euclid knew this. It soon will becoame larger than x. Read [wiki](https://en.wikipedia.org/wiki/Prime-counting_function) about $\pi(x)$ – 2017-01-23
-
0$$\sum_p \lfloor N /p \rfloor = \sum_p \sum_n 1_{pn \le N} =\sum_n \pi(N/n) $$ @miracle173 – 2017-01-23
-
0@miracle173 Corrected, thank you. – 2017-01-23
-
1I still don't think this is right. Besides that: I think the questions is much too broad, there is another site(http://hsm.stackexchange.com/) dedicated to history of mathematics – 2017-01-23