[proofplan] We identify Dirichlet characters modulo $q$ with the complex character group of the finite abelian group $G = (\mathbb{Z}/q\mathbb{Z})^\times$. Since $a$ and $n$ are both coprime to $q$, the summand becomes the value of each group character at the single group element $\bar n \bar a^{-1}$. The result is then the finite-group character orthogonality relation: the average of all characters is $1$ at the identity and $0$ away from the identity. We prove the needed separation fact for finite abelian groups directly, so no external character-theory theorem is being invoked. [/proofplan]

[step:Translate the Dirichlet character sum to a finite abelian group character sum]Let \begin{align*} G := (\mathbb{Z}/q\mathbb{Z})^\times \end{align*} be the multiplicative group of invertible residue classes modulo $q$. Its order is $|G|=\varphi(q)$. Let \begin{align*} \widehat G := \{\psi: G \to \mathbb{C}^\times : \psi(xy)=\psi(x)\psi(y) \text{ for all } x,y \in G\} \end{align*} be the group of complex-valued group characters of $G$. A Dirichlet character modulo $q$ is precisely a character $\psi \in \widehat G$, viewed as a function $\chi:\mathbb{Z}\to\mathbb{C}$ by setting $\chi(m)=\psi(\bar m)$ when $\gcd(m,q)=1$ and $\chi(m)=0$ when $\gcd(m,q)>1$. Since $\gcd(a,q)=\gcd(n,q)=1$, both $\bar a$ and $\bar n$ lie in $G$, and every corresponding Dirichlet character satisfies \begin{align*} \chi(a)=\psi(\bar a), \qquad \chi(n)=\psi(\bar n). \end{align*} For every $\psi\in \widehat G$, the value $\psi(\bar a)$ is a root of unity, because $\bar a$ has finite order in $G$. Hence $|\psi(\bar a)|=1$ and \begin{align*} \overline{\psi(\bar a)}=\psi(\bar a)^{-1}=\psi(\bar a^{-1}). \end{align*} Therefore \begin{align*} \overline{\chi(a)}\,\chi(n) = \overline{\psi(\bar a)}\,\psi(\bar n) = \psi(\bar a^{-1})\psi(\bar n) = \psi(\bar n \bar a^{-1}). \end{align*} Defining \begin{align*} g := \bar n \bar a^{-1} \in G, \end{align*} the desired average becomes \begin{align*} \frac{1}{\varphi(q)}\sum_{\chi \bmod q}\overline{\chi(a)}\,\chi(n) = \frac{1}{|G|}\sum_{\psi\in\widehat G}\psi(g). \end{align*}[/step]

[guided]The point of this step is to remove the arithmetic notation and expose the underlying group-theoretic statement. Since $a$ and $n$ are both coprime to $q$, their residue classes $\bar a$ and $\bar n$ are elements of the unit group \begin{align*} G := (\mathbb{Z}/q\mathbb{Z})^\times. \end{align*} A Dirichlet character modulo $q$ is a group character \begin{align*} \psi: G \to \mathbb{C}^\times \end{align*} extended to all integers by assigning the value $0$ on residue classes not coprime to $q$. Because $a$ and $n$ are units modulo $q$, no zero values occur in this proof. For $\psi\in\widehat G$, the value $\psi(\bar a)$ has finite multiplicative order, since $\bar a$ has finite order in the finite group $G$. Thus $\psi(\bar a)$ lies on the unit circle, and complex conjugation agrees with inversion: \begin{align*} \overline{\psi(\bar a)}=\psi(\bar a)^{-1}. \end{align*} Using the homomorphism property of $\psi$, we get \begin{align*} \overline{\chi(a)}\,\chi(n) = \overline{\psi(\bar a)}\,\psi(\bar n) = \psi(\bar a)^{-1}\psi(\bar n) = \psi(\bar a^{-1})\psi(\bar n) = \psi(\bar n\bar a^{-1}). \end{align*} Thus the whole Dirichlet character sum is the average of all group characters evaluated at the single element \begin{align*} g := \bar n\bar a^{-1}\in G. \end{align*} So it remains to prove the finite-group assertion \begin{align*} \frac{1}{|G|}\sum_{\psi\in\widehat G}\psi(g) = \begin{cases} 1, & g=e,\\ 0, & g\ne e. \end{cases} \end{align*}[/guided]

[step:Prove that finite abelian group characters separate nonidentity elements] [claim:Every nonidentity element of a finite abelian group is detected by a character] Let $A$ be a finite abelian group, and let $x\in A$ with $x\ne e_A$. Then there exists a character \begin{align*} \theta: A \to \mathbb{C}^\times \end{align*} such that $\theta(x)\ne 1$. [/claim] [proof] Let $m\in\mathbb{N}$ be the order of $x$, so $m>1$. Let $H:=\langle x\rangle\subset A$ be the cyclic subgroup generated by $x$. Define \begin{align*} \theta_0: H &\to \mathbb{C}^\times \\ x^k &\mapsto \exp(2\pi i k/m). \end{align*} This is well-defined because $x^k=x^\ell$ holds exactly when $k\equiv \ell \pmod m$, and the exponential above has period $m$. It is a group homomorphism, and \begin{align*} \theta_0(x)=\exp(2\pi i/m)\ne 1. \end{align*} It remains to extend $\theta_0$ from $H$ to all of $A$. Consider all pairs $(B,\eta)$ where $B$ is a subgroup of $A$ containing $H$ and \begin{align*} \eta:B\to\mathbb{C}^\times \end{align*} is a group homomorphism extending $\theta_0$. Since $A$ is finite, choose such a pair $(B,\eta)$ with $B$ maximal by inclusion. We show that $B=A$. Suppose not. Choose $y\in A\setminus B$. Let $r\in\mathbb{N}$ be the smallest positive integer such that $y^r\in B$; such an $r$ exists because $A$ is finite. Choose $\lambda\in\mathbb{C}^\times$ satisfying \begin{align*} \lambda^r=\eta(y^r), \end{align*} which exists because every nonzero complex number has an $r$-th root. Let $B' := \langle B,y\rangle$. Every element of $B'$ has a unique representation $b y^k$ with $b\in B$ and $0\le k<r$. Indeed, existence follows by reducing the exponent of $y$ modulo $r$, and uniqueness follows from the minimality of $r$: if $b y^k=b' y^\ell$ with $0\le k,\ell<r$, then $y^{k-\ell}\in B$, so $k=\ell$ and then $b=b'$. Define \begin{align*} \eta':B' &\to \mathbb{C}^\times \\ b y^k &\mapsto \eta(b)\lambda^k \end{align*} for $b\in B$ and $0\le k<r$. The uniqueness just proved makes $\eta'$ well-defined. If $b,d\in B$ and $0\le k,\ell<r$, write $k+\ell=sr+t$ with $s\in\{0,1\}$ and $0\le t<r$. Since $A$ is abelian, \begin{align*} (b y^k)(d y^\ell)=bd(y^r)^s y^t. \end{align*} Therefore \begin{align*} \eta'((b y^k)(d y^\ell)) &= \eta(bd(y^r)^s)\lambda^t \\ &= \eta(b)\eta(d)\eta(y^r)^s\lambda^t \\ &= \eta(b)\eta(d)\lambda^{rs+t} \\ &= \eta(b)\lambda^k\,\eta(d)\lambda^\ell \\ &= \eta'(b y^k)\eta'(d y^\ell). \end{align*} Thus $\eta'$ is a character on $B'$ extending $\eta$. Since $y\notin B$, we have $B\subsetneq B'$, contradicting the maximality of $B$. Hence $B=A$. The maximal extension $\eta:A\to\mathbb{C}^\times$ extends $\theta_0$, so \begin{align*} \eta(x)=\theta_0(x)=\exp(2\pi i/m)\ne 1. \end{align*} Taking $\theta:=\eta$ proves the claim. [/proof] Apply the claim with $A=G$. If $g\ne e_G$, there exists $\psi_0\in\widehat G$ with $\psi_0(g)\ne 1$. [/step]

[step:Count the characters of a finite abelian group] [claim:A finite abelian group has exactly as many complex characters as elements] Let $A$ be a finite abelian group, and let \begin{align*} \widehat A := \{\theta:A\to\mathbb{C}^\times : \theta(xy)=\theta(x)\theta(y) \text{ for all } x,y\in A\} \end{align*} be its complex character group. Then $|\widehat A|=|A|$. [/claim] [proof] We argue by induction on $|A|$. If $|A|=1$, then $A=\{e_A\}$ and the only character is the constant character $e_A\mapsto 1$, so $|\widehat A|=1=|A|$. Assume $|A|>1$ and that the assertion is known for all finite abelian groups of smaller order. Choose $x\in A$ with $x\ne e_A$, let $m\in\mathbb{N}$ be the order of $x$, and let $H:=\langle x\rangle\subset A$. Then $1<m\le |A|$. The cyclic group $H$ has exactly $m$ characters: for each integer $j\in\{0,1,\dots,m-1\}$, define \begin{align*} \theta_j:H &\to \mathbb{C}^\times \\ x^k &\mapsto \exp(2\pi i jk/m), \end{align*} and these are all characters because a character on $H$ is determined by its value on $x$, which must be an $m$-th root of unity. Define the restriction map \begin{align*} R:\widehat A &\to \widehat H \\ \theta &\mapsto \theta|_H. \end{align*} We prove that $R$ is surjective. Let $\theta_0:H\to\mathbb{C}^\times$ be a character. Consider all pairs $(B,\eta)$ where $B$ is a subgroup of $A$ containing $H$ and $\eta:B\to\mathbb{C}^\times$ is a character extending $\theta_0$. Since $A$ is finite, choose such a pair maximal by inclusion. If $B\ne A$, choose $y\in A\setminus B$, let $r\in\mathbb{N}$ be minimal with $y^r\in B$, choose $\lambda\in\mathbb{C}^\times$ with $\lambda^r=\eta(y^r)$, and define on $B':=\langle B,y\rangle$ the map \begin{align*} \eta':B' &\to \mathbb{C}^\times \\ b y^k &\mapsto \eta(b)\lambda^k, \end{align*} where $b\in B$ and $0\le k<r$. The same uniqueness and multiplication computation as in the preceding separation claim shows that $\eta'$ is a well-defined character extending $\eta$, contradicting maximality. Hence $B=A$, so $\theta_0$ extends to a character of $A$ and $R$ is surjective. The kernel of $R$ is \begin{align*} \ker R = \{\theta\in\widehat A : \theta(h)=1 \text{ for all } h\in H\}. \end{align*} Characters in $\ker R$ are exactly the characters that factor through the [quotient group](/page/Quotient%20Group) $A/H$: the bijection is \begin{align*} \widehat{A/H} &\to \ker R \\ \rho &\mapsto \bigl(a\mapsto \rho(aH)\bigr). \end{align*} The map is well-defined because $H$ is a subgroup of the abelian group $A$, hence $A/H$ is a finite abelian group. It is injective because every coset $aH$ is represented by some $a\in A$, and it is surjective because if $\theta\in\ker R$, then $\rho(aH):=\theta(a)$ is independent of the representative: whenever $aH=bH$, one has $b^{-1}a\in H$, and therefore \begin{align*} \theta(a)=\theta(b)\theta(b^{-1}a)=\theta(b). \end{align*} Thus $|\ker R|=|\widehat{A/H}|$. Since $R:\widehat A\to\widehat H$ is a surjective homomorphism of finite groups, the fiber-counting formula for a homomorphism gives \begin{align*} |\widehat A|=|\ker R|\,|\widehat H|. \end{align*} The group $A/H$ has smaller order than $A$, so the induction hypothesis gives $|\widehat{A/H}|=|A/H|$. Combining this with $|\widehat H|=|H|=m$, we obtain \begin{align*} |\widehat A| = |\widehat{A/H}|\,|\widehat H| = |A/H|\,|H| = |A|. \end{align*} This completes the induction. [/proof] Applying the claim with $A=G$ gives $|\widehat G|=|G|$. [/step]

[step:Evaluate the character average at the identity] If $g=e_G$, then every $\psi\in\widehat G$ satisfies $\psi(g)=\psi(e_G)=1$. Since the preceding step proved $|\widehat G|=|G|$, we get \begin{align*} \frac{1}{|G|}\sum_{\psi\in\widehat G}\psi(e_G) = \frac{1}{|G|}\sum_{\psi\in\widehat G}1 = \frac{|\widehat G|}{|G|} = 1. \end{align*} [/step]

[step:Force the character average to vanish away from the identity]Assume $g\ne e_G$. By the separation claim, choose $\psi_0\in\widehat G$ such that $\psi_0(g)\ne 1$. Define the character-sum function \begin{align*} S:G &\to \mathbb{C} \\ h &\mapsto \sum_{\psi\in\widehat G}\psi(h). \end{align*} Multiplication by $\psi_0$ permutes $\widehat G$, because the map \begin{align*} M_{\psi_0}:\widehat G &\to \widehat G \\ \psi &\mapsto \psi_0\psi \end{align*} has inverse $M_{\psi_0^{-1}}$. Therefore \begin{align*} S(g) &= \sum_{\psi\in\widehat G}(\psi_0\psi)(g) \\ &= \sum_{\psi\in\widehat G}\psi_0(g)\psi(g) \\ &= \psi_0(g)S(g). \end{align*} Since $\psi_0(g)\ne 1$, subtracting the right-hand side gives \begin{align*} (1-\psi_0(g))S(g)=0, \end{align*} and hence $S(g)=0$. Thus \begin{align*} \frac{1}{|G|}\sum_{\psi\in\widehat G}\psi(g)=0 \end{align*} whenever $g\ne e_G$.[/step]

[guided]Here is the cancellation mechanism in full detail. We know $g\ne e_G$, so the previous step gives a character \begin{align*} \psi_0:G\to\mathbb{C}^\times \end{align*} with $\psi_0(g)\ne 1$. Define the character-sum function \begin{align*} S:G &\to \mathbb{C} \\ h &\mapsto \sum_{\psi\in\widehat G}\psi(h). \end{align*} The key idea is to evaluate this function at $g$ and multiply every character in the sum by the fixed character $\psi_0$. This does not change the set being summed over: the map \begin{align*} M_{\psi_0}:\widehat G &\to \widehat G \\ \psi &\mapsto \psi_0\psi \end{align*} is a bijection, with inverse $\psi\mapsto \psi_0^{-1}\psi$. Because this relabelling is a bijection, the sum is unchanged: \begin{align*} S(g) = \sum_{\psi\in\widehat G}(\psi_0\psi)(g). \end{align*} Now evaluate the product character at $g$: \begin{align*} (\psi_0\psi)(g)=\psi_0(g)\psi(g). \end{align*} Since $\psi_0(g)$ is independent of $\psi$, it factors out: \begin{align*} S(g) = \sum_{\psi\in\widehat G}\psi_0(g)\psi(g) = \psi_0(g)\sum_{\psi\in\widehat G}\psi(g) = \psi_0(g)S(g). \end{align*} Thus \begin{align*} (1-\psi_0(g))S(g)=0. \end{align*} The factor $1-\psi_0(g)$ is nonzero by construction, so $S(g)=0$. This is the orthogonality cancellation: away from the identity, the character values cancel exactly.[/guided]

[step:Convert the group identity condition back to the congruence condition] By definition, \begin{align*} g=e_G \iff \bar n \bar a^{-1}=1 \iff \bar n=\bar a \iff n\equiv a \pmod q. \end{align*} Combining this equivalence with the preceding two steps gives \begin{align*} \frac{1}{\varphi(q)}\sum_{\chi \bmod q}\overline{\chi(a)}\,\chi(n) = \frac{1}{|G|}\sum_{\psi\in\widehat G}\psi(g) = \begin{cases} 1, & n \equiv a \pmod q,\\ 0, & n \not\equiv a \pmod q. \end{cases} \end{align*} This is exactly the asserted orthogonality relation for Dirichlet characters modulo $q$. [/step]