Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ the number of primitive $n$th roots of unity with negative real part. Then $\varphi(n)=R(n)+L(n)$ for $n\ne 4$. I am particularly interested in (references to) an explicit formula for the difference function $D(n):=R(n)-L(n)$. I was surprised to observe (and then prove) that the values of $D(n)$ are zero, or plus or minus a power of 2. For example, $$ D(3)=-2,\quad D(6)=2,\quad D(8)=0,\quad D(21)=4,\quad D(42)=-4. $$ Three applications of the formula are: $\limsup_{n\to\infty} D(n)=\infty$, $\lim_{n\to\infty}D(n)/\varphi(n)=0$, and $\frac{\varphi(n)}{3}\leqslant L(n)\leqslant\frac{2\varphi(n)}{3}$ for $n>6$.