Gauss sums are analogous to the Gamma function:
Fix a complex number $s$ with real part $>0$. Then we have a multiplicative character $\chi_s :\mathbb R^{\times}_{>0} \to \mathbb C^\times$ given by $x \mapsto x^s$. We also have an additive character $\lambda: \mathbb R \to \mathbb C^\times$ given by $x \mapsto e^{-x}$. The inner product of these two characters over $\mathbb R^{\times}_{>0}$, $$\langle \chi_s, \lambda \rangle : = \int_0^\infty x^s e^{-x} \frac{dx}{x}$$ is the Gamma function.
Remark that $dx/x$ is the Haar measure on $\mathbb R^{\times}_{>0}$.
For Gauss sums, say over the finite field $\mathbb F_p$, we take the inner product of a Dirichlet character $\chi: \mathbb F_p^\times \to \mathbb C^\times$ with the additive character $\lambda : \mathbb F_p \to \mathbb C^\times$ given by $x \mapsto e^{2\pi i x/p}$:
$$\eta(\chi) = \sum_{x \in \mathbb F_p^\times} \chi(x) \lambda(x).$$
With this notation, the Gamma function 'is' $\eta(\chi_s)$.
There are many analogies between Gauss sums and the Gamma function (Jacobi sums vs Euler beta function, Gauss multiplication formula vs Hasse–Davenport product relation...).
But I haven't seen a Gauss sum equivalent of the relation $\eta(\chi_{s+1}) = s\eta(\chi_s)$. Is there one, and if so what could it be?
