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Using a result of Erdos as in this question

An upper bound for log 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.

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up vote 4 down vote accepted

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.

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Better: – Charles Jul 29 '11 at 3:23

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).

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See also… – 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… – 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

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