The Wang--Sun Sum over Affine Derangements

Let $n$ be even, let $\zeta$ be a primitive $n$-th root of unity, and let ${\mathrm{\Delta }}_n$ be the set of derangements in the affine group $\overrightarrow{GL}(\mathbb{Z}\mathrm{/}n\mathbb{Z})$. We evaluate the Wang–Sun type sum $$S_n=\sum _{\pi \in {\mathrm{\Delta }}_n}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(\pi )\prod _{x\in \mathbb{Z}\mathrm{/}n\mathbb{Z}}\frac{1+{\zeta }^{x-\pi (x)}}{1-{\zeta }^{x-\pi (x)}}\mathrm{.}$$ The main step is an arithmetic grouping: for an affine map $\pi (x)=vx+u$, the product depends only on $\sigma =\gcd (1-v,n)$ and on the residue class of $u$ modulo $\sigma$. After summing the translation signs, the full sum becomes a sum over even divisors with explicit local multiplicities. The only remaining analytic blocks are alternating even cotangent power sums; a recent formula of Liu and Xin then gives a closed expression in Bernoulli polynomials and universal coefficients. Moreover, using results by Cvijović and Klinowski we can evaluate the cotangent power sums using exclusively rational arithmetic. This supplies a general formula for the full affine sum whose initial values were computed using the definition only in an earlier work. The formula also lets us prove some general patterns of the sum: it has integral value for powers of $2$, prime denominator for $2p$ for primes $p>3$ and it has alternating signs.