I have attached a proposition whose proof I don't understand at two points. Here are my questions:
- Why do we have $|a_{0n}+\theta_{1}a_{1n}+\dots+\theta_{k}a_{kn}|<(\rho-\varepsilon)^{-n}$ for sufficiently large $n$?
- Why do we have $[1,...,n]<e^{(1+\varepsilon)n}$ by the prime number theorem?
Thank you very much.
1) This follows immediately from the characterization of the radius of convergence of an arbitrary power series $\sum_{n=0}^\infty b_n z^n$ as $\rho^{-1} = \limsup_{n\to\infty} |b_n|^{1/n}$.
2) One of the equivalent formulations of the Prime Number Theorem is that the Chebyshev function $\psi(x)$ is asymptotic to $x$, i.e. $\lim_{x\to\infty} \psi(x)/x = 1$. If you read the definition of $\psi(x)$ carefully, you will find that $\exp(\psi(n))$ is exactly equal to the LCM $[1,2,\ldots,n]$, by considering how many times the term $\log p$ appears in $\psi(n)$ for each prime $p$. It then follows immediately that $[1,2,\ldots,n] \le \exp((1+\epsilon)n)$ for sufficiently large $n$.