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

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.

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.

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).

• – lhf Jul 28 '11 at 18:14
• Thanks for posting that link to Rosser's paper. I have a copy from when I was a math undergraduate, but it's nice to have an electronic copy. It's a little dated (done in 1961, refers to computations done on an IBM 650, ...), so I wonder what has been done in the last 50 years. – marty cohen Jul 31 '11 at 23:15
• @marty: I have converted your answer to a comment on lhf's post. Because you do not have 50 reputation points yet, you can only comment on your own questions and answers. So, you didn't do anything wrong; the "add comment" button will only appear for you once you gain 50 points. Here is an explanation of reputation points. – Zev Chonoles Jul 31 '11 at 23:29
• @marty, I've added a link to some recent work. – lhf Aug 1 '11 at 1:13
• Also take a look at math.stackexchange.com/questions/59258/… – user14947 Sep 24 '11 at 23:08

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