[proofplan] We apply [Convergence of Dirichlet Series](/theorems/1616) to the series $L(\chi, s) = \sum_{n \geq 1} \chi(n) n^{-s}$ with $a_n = \chi(n)$. The hypothesis of Theorem 1616 requires that the partial sums $T(N) = \sum_{n=1}^N \chi(n)$ grow no faster than $N^r$ for some $r$; the conclusion is holomorphy on $\{\operatorname{Re}(s) > r\}$. To obtain holomorphy on $\{\operatorname{Re}(s) > 0\}$, we must show $T(N) = O(1)$, i.e., $r = 0$ works. This follows from the orthogonality relation for characters of $(\mathbb{Z}/D\mathbb{Z})^\times$ applied to the non-trivial character $\omega$ corresponding to $\chi$: for every complete residue system modulo $D$, the sum of $\chi$ over that system vanishes, so the partial sum $T(N)$ is bounded by the sum of $\chi$ over at most one incomplete block. [/proofplan] [step:Identify $\chi$ with a character of $(\mathbb{Z}/D\mathbb{Z})^\times$ and recall orthogonality] A Dirichlet character $\chi$ of modulus $D$ is a function $\chi : \mathbb{Z} \to \mathbb{C}$ that is: - completely multiplicative: $\chi(mn) = \chi(m)\chi(n)$ for all $m, n$; - periodic with period $D$: $\chi(n + D) = \chi(n)$; - zero exactly on $n$ with $\gcd(n, D) > 1$. Equivalently, $\chi$ is induced from a group homomorphism \begin{align*} \omega: (\mathbb{Z}/D\mathbb{Z})^\times &\to \mathbb{C}^\times, \end{align*} with $\chi(n) = \omega(n + D\mathbb{Z})$ when $\gcd(n, D) = 1$ and $\chi(n) = 0$ otherwise. The character $\chi$ is called **trivial** (also: principal) if $\omega$ is the constant function $1$. By hypothesis, $\chi$ is non-trivial, so $\omega$ is a non-trivial character. [claim:Orthogonality Sum] For any non-trivial character $\omega$ of a finite abelian group $G$, \begin{align*} \sum_{g \in G} \omega(g) &= 0. \end{align*} [/claim] [proof] Since $\omega$ is non-trivial, there exists $h \in G$ with $\omega(h) \neq 1$. Left-multiplying the sum variable $g$ by $h$ is a bijection $G \to G$ (group translation): \begin{align*} \sum_{g \in G} \omega(g) = \sum_{g \in G} \omega(hg) = \omega(h) \sum_{g \in G} \omega(g). \end{align*} Hence $(\omega(h) - 1) \sum_{g \in G} \omega(g) = 0$, and since $\omega(h) - 1 \neq 0$, $\sum_{g \in G} \omega(g) = 0$. [/proof] Applying the claim to $G = (\mathbb{Z}/D\mathbb{Z})^\times$ and $\omega$ the non-trivial character corresponding to $\chi$: \begin{align*} \sum_{g \in (\mathbb{Z}/D\mathbb{Z})^\times} \omega(g) &= 0. \end{align*} [guided] **Dirichlet characters as group characters.** A Dirichlet character $\chi$ of modulus $D$ is the pull-back of a group character $\omega : (\mathbb{Z}/D\mathbb{Z})^\times \to \mathbb{C}^\times$ through the projection $\mathbb{Z} \to \mathbb{Z}/D\mathbb{Z}$, extended by zero on non-coprime integers. This is the link between analytic and algebraic objects: $\omega$ lives on a finite group, $\chi$ lives on the integers. **Non-triviality.** The trivial character has $\omega(g) = 1$ for all $g$, and $\chi = \mathbb{1}_{\gcd(\cdot, D) = 1}$ (the principal Dirichlet character). We exclude this case by hypothesis, so $\omega$ is not constantly $1$: some $h \in (\mathbb{Z}/D\mathbb{Z})^\times$ has $\omega(h) \neq 1$. **Orthogonality.** For a non-trivial character of a finite abelian group, the full sum vanishes. The proof is a classical translation argument: the sum is invariant under the group action (reparametrise $g \to hg$), giving $(\omega(h) - 1) \sum_g \omega(g) = 0$, and if $\omega(h) \neq 1$ then $\sum_g \omega(g) = 0$. **Why only non-trivial?** For the trivial character, $\sum_g \omega(g) = \sum_g 1 = |G| \neq 0$, so the partial sums of $\mathbb{1}_{\gcd(\cdot, D) = 1}$ grow linearly, and the corresponding $L$-function $L(\chi_1, s) = \zeta(s) \prod_{p \mid D}(1 - p^{-s})$ has a pole at $s = 1$, not holomorphy. [/guided] [/step] [step:Use orthogonality to show $T(N) = O(1)$] Fix $N \geq 1$ and write $N = aD + b$ with $a \in \mathbb{Z}_{\geq 0}$ and $0 \leq b < D$. The partial sum of $\chi$ decomposes as \begin{align*} T(N) &= \sum_{n=1}^N \chi(n) = \sum_{k=0}^{a-1} \sum_{n = kD + 1}^{(k+1)D} \chi(n) + \sum_{n = aD + 1}^{aD + b} \chi(n). \end{align*} For each complete block $n \in \{kD + 1, \ldots, (k+1)D\}$, periodicity gives $\chi(kD + j) = \chi(j)$ for $j = 1, \ldots, D$: \begin{align*} \sum_{n = kD + 1}^{(k+1)D} \chi(n) &= \sum_{j = 1}^{D} \chi(j). \end{align*} The sum $\sum_{j=1}^D \chi(j)$ is a sum over one complete residue system modulo $D$. Splitting by $\gcd(j, D)$: - If $\gcd(j, D) > 1$, $\chi(j) = 0$. - If $\gcd(j, D) = 1$, $\chi(j) = \omega(j + D\mathbb{Z})$. Hence \begin{align*} \sum_{j=1}^D \chi(j) &= \sum_{\substack{j = 1 \\ \gcd(j, D) = 1}}^D \omega(j + D\mathbb{Z}) = \sum_{g \in (\mathbb{Z}/D\mathbb{Z})^\times} \omega(g) = 0, \end{align*} by the orthogonality identity from Step 1. Therefore each complete block contributes zero, and \begin{align*} T(N) &= \sum_{n = aD + 1}^{aD + b} \chi(n). \end{align*} The remaining sum contains at most $b < D$ terms, each with $|\chi(n)| \leq 1$ (since $\chi(n)$ is either $0$ or a root of unity). Hence \begin{align*} |T(N)| &\leq b < D, \end{align*} a bound independent of $N$. In particular, $T(N) = O(1) = O(N^0)$. [guided] **Periodicity partitions the partial sum.** Writing $N = aD + b$ with $0 \leq b < D$, we split the sum into $a$ complete blocks of length $D$ plus a short tail of length $b$. Each complete block is the sum of $\chi$ over a complete residue system modulo $D$, and by periodicity these blocks are identical copies of $\sum_{j=1}^D \chi(j)$. **Each complete block sums to zero.** The sum over one period is \begin{align*} \sum_{j=1}^D \chi(j) = \sum_{\substack{j = 1 \\ \gcd(j, D) = 1}}^D \chi(j) + \sum_{\substack{j = 1 \\ \gcd(j, D) > 1}}^D \chi(j) = \sum_{g \in (\mathbb{Z}/D\mathbb{Z})^\times} \omega(g) + 0, \end{align*} using that $\chi$ vanishes exactly on the non-coprime integers. The first sum is the full character sum over $(\mathbb{Z}/D\mathbb{Z})^\times$, which equals zero by the orthogonality of Step 1 applied to the non-trivial character $\omega$. **Tail estimate.** The remaining sum has at most $b \leq D - 1 < D$ terms, each of absolute value at most $1$ (Dirichlet characters take values in $\{0\} \cup \mu_k$ for some $k$, where $\mu_k$ is the set of $k$-th roots of unity, all of modulus $1$). Hence $|T(N)| \leq D - 1 < D$ for every $N$. **Conclusion.** The partial sum $T(N)$ is bounded uniformly in $N$ by $D - 1$. In the asymptotic language of [Theorem 1616](/theorems/1616), $T(N) = O(N^0) = O(1)$, which is the case $r = 0$. [/guided] [/step] [step:Apply Theorem 1616 to conclude holomorphy on $\{\operatorname{Re}(s) > 0\}$] The Dirichlet series associated to $\chi$ is \begin{align*} L(\chi, s) &= \sum_{n = 1}^\infty \chi(n) n^{-s}, \end{align*} with coefficients $a_n = \chi(n)$. By Step 2, the partial sums $T(N) = \sum_{n=1}^N \chi(n) = O(N^0) = O(1)$. By [Convergence of Dirichlet Series](/theorems/1616): the hypothesis is that there exists $r \in \mathbb{R}$ with $a_1 + \cdots + a_N = O(N^r)$. Taking $r = 0$, this hypothesis is satisfied. The conclusion of Theorem 1616 gives that the series converges for every $s$ with $\operatorname{Re}(s) > 0$ and defines a holomorphic function on $\{\operatorname{Re}(s) > 0\}$. This establishes the claim: $L(\chi, s)$ is holomorphic for $\operatorname{Re}(s) > 0$. [guided] **Invoking Theorem 1616.** [Convergence of Dirichlet Series](/theorems/1616) is the following bootstrap statement: if the partial sums of $a_n$ are $O(N^r)$, then $\sum a_n n^{-s}$ converges and is holomorphic on $\{\operatorname{Re}(s) > r\}$. **Hypothesis verification.** We need $a_1 + \cdots + a_N = O(N^r)$ for some real $r$. By Step 2, we have $T(N) = O(1) = O(N^0)$, so we may take $r = 0$. **Conclusion.** The conclusion of Theorem 1616 with $r = 0$ states that the Dirichlet series $\sum \chi(n) n^{-s}$ converges for all $s$ with $\operatorname{Re}(s) > 0$, and defines a holomorphic function on $\{\operatorname{Re}(s) > 0\}$. **Meaning.** The non-trivial $L$-function $L(\chi, s)$ extends to a holomorphic function on the half-plane $\{\operatorname{Re}(s) > 0\}$ — this half-plane includes $s = 1$, which is important because the non-vanishing of $L(\chi, 1)$ is the next step in applications such as [Dirichlet's Theorem on Primes in Arithmetic Progressions](/theorems/1625). **Contrast with the trivial case.** For the trivial character, $T(N) \sim \varphi(D)N/D$ grows linearly, so we only get $r = 1$: holomorphy on $\{\operatorname{Re}(s) > 1\}$. At $s = 1$ there is in fact a pole, matching the fact that the trivial $L$-function is essentially the Riemann zeta function up to finitely many Euler factors. [/guided] [/step]