Let $N\geq 2$ be a natural number, then let $\chi: (\mathbb{Z}/N\mathbb{Z})^\times \to \{\pm 1 \}$ be a quadratic character. Consider the Gauss sum $$\tau(1,\chi) = \sum_{x\in (\mathbb{Z}/N\mathbb{Z})^\times} \chi(x)e^{2\pi i x/N}. $$
Most books I see touch on this subject (e.g., Ireland-Rosen and Lang) show that $|\tau(1,\chi)|=\sqrt{N}$ for $N$ prime.
Does this generalize to $N$ composite? These notes here seem to argue that it does but i'm not sure on the details: http://www.math.leidenuniv.nl/~evertse/ant13-4.pdf. See Theorem 4.17. Is there an original reference for this?
Is there a formula for the argument of $\tau(1,\chi)$?
Only for the primitive quadratic character (not of the form $\psi(x)1_{\gcd(x,N)=1}$ for $\psi$ a quadratic character $\bmod d$). Then the discrete Fourier transform of $\chi$ is $\tau(\chi,k)=\overline{\chi(k)} \tau(\chi,1)$ (immediate for $\gcd(k,N)=1$, use primitivity to prove $\tau(\chi,d)=0$ for $d| N$)
thus $$\phi(N) |\tau(\chi,1)|^2 =\sum_{k=1}^N |\tau(\chi,k)|^2=N \sum_n |\chi(n)|^2 = N\phi(N)$$
And since $\overline{\tau(\chi,1)} = \chi(-1)\tau(\chi,1)$ we find $$\tau(\chi,1)^2 = \chi(-1) N$$ The sign of $\tau(\chi,1)=\pm \sqrt{\chi(-1) N}$ is more complicated to find, search "sign of Gauss sum".
Also note that for $\chi_1,\chi_2$ any Dirichlet characters $\bmod N_1,N_2$ such that $aN_1+bN_2=1$ and $\chi=\chi_1\chi_2$ then $$\tau(\chi,1) = \sum_{x\bmod N_1,y\bmod N_2} \chi(bN_2x+aN_1 y)e^{2i \pi (aN_2 x+bN_1 y)/(N_1N_2)}\\ =\sum_{x\bmod N_1,y\bmod N_2} \chi_1(x)\chi_2(y)e^{2i \pi x/N_1}e^{2i\pi y/N_2}=\tau(\chi_1,1)\tau(\chi_2,1)$$