Roots of unity

Let's start with a preliminary remark: If $w$ is a primitive $n$ th root of unity, where $n$ is odd, then $1-z^n=\prod_{b=1}^{n}(w^b-z)$ . Plugging in $z=-1$ , we find that $\prod_{b=1}^{n}(1+w^b)=2$ .

Now let $w=e^{2\pi i a/2015}$ and $d=\gcd(a,2015)$ . Note that $w$ is a primitive $(2015/d)$ th root of unity, so that $\prod_{b=1}^{2015/d}(1+w^b)=2$ by the preliminary remark. Running through this product $d$ times, we find that $\prod_{b=1}^{2015}(1+w^b)=2^d=2^{\gcd(2015,a)}$ . Thus $\prod_{a=1}^{2015}\prod_{b=1}^{2015}(1+w^b)=2^{\sum_{a=1}^{2015}\gcd(2015,a)}$ .

The logarithm is $\sum_{a=1}^{2015}\gcd(a,2015)=\prod_{p|2015}(2p-1)=9*25*61=\boxed{13725}$ for the three prime factors $p=5,11,31$ of $2015$ (see this )

This is from the recent 2015 Putnam: http://www.artofproblemsolving.com/community/c7t441f7h1171025 putnam 2015_a3 .