Character sum

Definition

Let $q \ge 1$ be an integer, and let $\chi$ be a Dirichlet character modulo $q$ . Let $M$ be an integer and $N \ge 1$ . The character sum of $\chi$ over the interval $(M, M + N]$ is $S_\chi(M, N) = \sum_{n = M + 1}^{M + N} \chi(n).$ More generally, a character sum is a sum $\sum_{n \in A} \chi(f(n))$ over a finite set $A$ of integers, where $f$ is a polynomial with integer coefficients.

Remarks

Example: let $\chi$ be the Legendre symbol modulo $5$ , so $\chi(1) = \chi(4) = 1$ , $\chi(2) = \chi(3) = -1$ and $\chi(5) = 0$ . Then $S_\chi(0, 3) = 1 - 1 - 1 = -1$ and $S_\chi(0, 5) = 0$ . Non-example: the sum $\sum_{n \le N} \mu(n)$ of the Möbius function is not a character sum, since $\mu$ is not a Dirichlet character: $\mu(4) = 0$ but $\mu(2)\mu(2) = 1$ , so $\mu$ is not completely multiplicative. Orthogonality gives the complete sum: $S_\chi(M, q) = 0$ if $\chi$ is not principal, and $S_\chi(M, q) = \varphi(q)$ if $\chi$ is principal, where $\varphi$ is Euler's totient function. Thus $\abs{S_\chi(M, N)} \le \varphi(q)$ for every nonprincipal $\chi$ . The Pólya–Vinogradov inequality improves this to $\abs{S_\chi(M, N)} \le C \sqrt{q} \log q$ for nonprincipal $\chi$ , with an absolute constant $C$ .

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Dirichlet character concept
Character sum definition

Tag	`4JWHKNGS4Y`
URI	`m2://4JWHKNGS4Y`
Kind	definition
Writing	W0 stub
MSC	11L40, 11L10
Defines	Character sum
Uses	Dirichlet character
Changed	2 November 2026

Character sum

Definition

Remarks

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