Thread: PNT

**Cairo** · Jul 20th 2009, 11:04 PM

I have managed to do part (a) and used Abel's Identity to prove part (b), but I don't know how to proceed with part (c).

Any help would be great here.

**chiph588@** · Jul 21st 2009, 02:08 PM

Ok, from part b we're given $g(x) = f(x) \log(x) - \int_{2}^{x} \frac{f(t)}{t} dt$ .

Now let $x$ tend to infinity on both sides.
You need to show $\int_{2}^{x} \frac{f(t)}{t} dt = o(x)$ as $x \to \infty$ . Then we know $f(x) \sim \frac{1}{4}\pi(x) \sim \frac{1}{4} \frac{x}{\log(x)}$ and we're done.

Notice since $f(x) \sim \frac{1}{4}\pi(x) \;,\;\; \frac{f(x)}{x} \sim \frac{1}{4}\frac{\pi(x)}{x} \sim \frac{1}{4}\frac{1}{\log(x)}$ . So $\int_{2}^{x} \frac{f(t)}{t} dt \sim \frac{1}{4}\int_{2}^{x}\frac{1}{\log(t)} dt$ . This now becomes easier to verify as $o(x)$ .
See here http://en.wikipedia.org/wiki/Logarit...otic_expansion.

**Cairo** · Aug 9th 2009, 12:25 AM

I've searched the net and I still can't seem to verify that the integral is little oh of x.

Can anybody help here?

Thread: PNT

LinkBack

Thread Tools

Display

PNT

Search Tags