non-trivial upper bound for the number of primes less or equal to n

Question

Using a result of Erdos as in this question

An upper bound for $\log \operatorname{rad}(n!)$

one can show that

$\sum_{p\leq n} \log p \leq \log(4) n$ for any positive integer $n$.

Trivially, $\sum_{p\leq n} 1 \leq n$.

Are there any other non-trivial upper bounds for $\sum_{p\leq n} 1$?

Note that I'm asking for upper bounds and not just asymptotic behaviour. Moreover, this is probably connected to the prime number theorem.

Answer 1

Many proofs of the prime number theorem involve some bounds. I'm familiar with a result of Pierre Dusart, stating that for all x, $\pi(x) \leq \frac{x}{\log x}(1 + \frac{1.2762}{\log x})$.

He was actually more proud of his lower bound. His paper is here.

Answer 2

Your sum is just $\pi(n)$, the number of primes less than or equal to $n$. This is the subject of the prime number theorem

Answer 3

See Explicit bounds for some functions of prime numbers by Rosser (1941, MR0003018). Among other results, there is $$\frac x{\log x+2} < \pi(x) < \frac x{\log x-4},\quad\mbox{for } x\geq 55$$

Similar explicit bounds can be found in Approximate formulas for some functions of prime numbers by Rosser and Schoenfeld (1962, MR0137689).

For a sample of recent work, see Short effective intervals containing primes by Ramaré and Saouter (2004, MR1950435).