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.

See also https://math.stackexchange.com/questions/28540 – Watson Dec 27 '19 at 17:31 — Watson, Dec 27 '19 at 17:31

score 5 · Accepted Answer · answered Jul 28 '11 at 17:31

5

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.

answered Jul 28 '11 at 17:31

davidlowryduda

91,687

Better: http://arxiv.org/abs/1002.0442 – Charles Jul 29 '11 at 03:23

lhf · Answer 2 · 2011-08-01T01:15:46.550

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).

edited Aug 01 '11 at 01:15

answered Jul 28 '11 at 18:01

lhf

216,483

See also http://math.stackexchange.com/questions/16085/is-p-n-pin-4n-where-p-n-is-the-largest-prime-leq-n – 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 01 '11 at 01:13
Also take a look at http://math.stackexchange.com/questions/59258/lower-bound-for-the-prime-number-function – Sep 24 '11 at 23:08

score 2 · Answer 3 · answered Jul 28 '11 at 17:30

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

answered Jul 28 '11 at 17:30

Ross Millikan

374,822

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

3 Answers3

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