Thread: PNT

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.

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

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

Notice since $\displaystyle 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 $\displaystyle \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 $\displaystyle o(x) $.
See here http://en.wikipedia.org/wiki/Logarit...otic_expansion.

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?