Exercise 19, § 11

a) Assume the prime number theorem; if $\pi(x)$ is the number of primes $p \leq x$, then $\pi(x) = x / \log x$ as $x$ tends to $+\infty$. Deduce that for every odd integer $m$, there exists an increasing sequence of prime numbers $p_1 < p_2 < \ldots < p_m$ such that $p_m < p_1 + p_2$ (consider the prime numbers between $x$ and $3x/2$). b) With the finite sequence of prime numbers $p_j$ ($1 \leq j \leq m$) determined as in a), put $n = p_1 p_2 \ldots p_m$. Show that in the cyclotomic polynomial $\Phi_n(X)$ the terms of degree $< p_m + 1$ are the same as those of the polynomial

Deduce that the coefficient of $X^{p_m}$ in $\Phi_n(X)$ is $1 - m$ (« I. Schur's theorem »).