I What is the inequality for prime numbers in the Prime Number Theorem proof?

[PsychonautQQ]

I believe this is probably a high level undergraduate question, but i could easily be underestimating it and it's actually quite a bit higher than that.

I'm reading the Prime number theorem wikipedia page and I'm in part 4 under Proof sketch where sometime down they give in inequality:

x is a natural number, p is for prime's obviously: $\sum_{x^{1-\epsilon }\geq p\geq x}^{} \log p \geq \sum_{p\leq x}^{} logx$

Where epsilon is any value greater than O. (O is some special value that they use in computer science a lot apparently, I might need to understand this value better to understand this inequality, I'm not sure.

Can somebody help me understand this inequality? C'mon I know there are some really smart people here!

 

[Buzz Bloom]

O is some special value that they use in computer science a lot apparently

Hi PsychonautQQ:

I am not into number theory, but I am almost certain that the "O" is zero, i.e., "0".

Regards,
Buzz

 

[fresh_42]

Hi PsychonautQQ:

I am not into number theory, but I am almost certain that the "O" is zero, i.e., "0".

Regards,
Buzz

My guess is $O(something)$, e.g. $O(1)\, , \,O(\log x) \, , \, O(\log \log x)$ or even $o(1)$. In any case I assume there is more than one typo in it, e.g. the $\log x$ in a summation over $p$ looks suspicious.

 

[mfb]

Here is a link to the article, the inequality discussed is the 6th formula in this section.
"for any ε > 0" is a comparison with the number 0. The big O notation (with the letter O) is mentioned because it is used later in the same line.

 

[Buzz Bloom]

Here is a link to the article,

I found the equation in post #1 in the the linked article, although the post #1 version seems to have omitted a detail. It seems clear that the intent is

ε > zero​

and

O(x^1-ε)​

is using the "big O" notation.

Regards,
Buzz

 

[mfb]

I thought it is the 6th formula there, but the right hand side is not identical.

 

[PsychonautQQ]

Wow, I am so smart for confusing 0 and O haha.

I still don't understand the inequality though, can somebody explain why it makes sense intuitively or link a proof to me?

 

[mfb]

With the correct inequality from the Wikipedia article: We know $x^{1-\epsilon} < p$. Take the logarithm on both sides and you get the inequality used in the article.

 

[PsychonautQQ]

With the correct inequality from the Wikipedia article: We know $x^{1-\epsilon} < p$. Take the logarithm on both sides and you get the inequality used in the article.

And we know that x^(1-epsilon) < p because of the domain which is being summed over? p is required to be bigger?

 

[mfb]

We know it because the sum only runs over p satisfying this condition, right.