Exercise 1.1.1.
For \(\epsilon > 0\text{,}\) \(\lim_{n \to \infty} \frac{(1+\epsilon)n}{\log((1+\epsilon)n)} - \frac{n}{\log n} = \infty \text{.}\)
Section 1.1 Prime Number Theorem
Write \(\pi(x)\) for the number of primes less than or equal to \(x\text{.}\) The Prime Number Theorem (PNT) asserts that
\begin{equation}
\pi(x) \sim \frac{x}{\log x} \quad \text{as } x \to \infty \text{,}\tag{1.1.1}
\end{equation}
that is,
\begin{equation}
\lim_{x \to \infty} \frac{\pi(x)}{x/\log x} = 1 \text{.}\tag{1.1.2}
\end{equation}
Here \(\log\) is the natural logarithm.
The Prime Number Theorem was proved in 1896-97 by Hadamard and by de la Vallée Poussin independently.
A consequence of the PNT is that for any \(\epsilon > 0\) there exists \(N(\epsilon)\) such that for all \(n \geq N(\epsilon)\text{,}\) there exists a prime \(p\) such that \(n \lt p \leq (1+\epsilon)n\text{.}\)
Why does PNT imply this? The number of primes in the interval \((n , (1+\epsilon)n] \) is equal to
\begin{equation*}
\pi((1+\epsilon)n) - \pi(n)
\end{equation*}
which is asymptotic to
\begin{equation*}
\frac{(1+\epsilon)n}{\log((1+\epsilon)n)} - \frac{n}{\log n} \text{,}
\end{equation*}
which goes to infinity as \(n \to \infty\text{.}\)
Exercise 1.1.2.
If \(f(x)\text{,}\) \(g(x)\text{,}\) \(a(x)\text{,}\) \(b(x)\) are positive-valued functions and \(f \sim a\text{,}\) \(g \sim b\text{,}\) then \(f+g \sim a+b\text{.}\) State and prove suitable conditions for \(f-g \sim a-b\text{.}\)
Bertrand’s postulate is relatively weak (\(\epsilon = 1\)) but on the other hand it is claimed for all \(n \geq 1\text{,}\) not just \(n \geq \) some \(N(1)\text{.}\)
