Montgomery-Vaughan: sum_{q <= R} mu^2(q)/phi(q) >= log R

For a real number $R \ge 0$ put

the sum being over the positive integers $q$ at most $R$, with $\mu$ the Möbius function and $\varphi$ Euler's totient. Then

Only the squarefree $q$ contribute, and for those $\mu^{2}(q)=1$, so $G(R)=\sum_{q\le R,\ q\ \text{squarefree}}1/\varphi(q)$.

This is the elementary half of the estimate of van Lint and Richert, sharpened by Montgomery and Vaughan to $G(R)\ge\log R+1.07$ for $R\ge6$. It is the input that converts the averaging over moduli in the local $L^{2}$ estimate (Tao's Lemma 4.6) into the saving of $\log x/\log R$ over the global $L^{2}$ estimate of Lemma 4.5, and hence it underlies Corollary 4.7, the upper bound on the major-arc $L^{2}$ mass. The bound is sharp in order: $G(R)\sim\log R$ as $R\to\infty$.

Formalization Note The sum ranges over the integers from $1$ to $\lfloor R\rfloor$, and $\mu$ takes integer values whose square is cast to a real number. No lower bound on $R$ is needed: for $R<1$ the logarithm is negative while the sum is non-negative, and for $R=0$ the convention $\log 0=0$ still leaves the inequality true with an empty sum.