The Pólya-Vinogradov Inequality

Abstract

Let χ : Z → C be a primitive Dirichlet character modulo m. χ being a Dirichlet character modulo m means that χ(kn) = χ(k)χ(n) for all k, n, that χ(n + m) = χ(n) for all n, and that if gcd(n, m) > 1 then χ(n) = 0. χ being primitive means that the conductor of χ is m. The conductor of χ is the smallest defining modulus of χ. If m is a divisor of m, m is said to be a defining modulus of χ if gcd(n, m) = 1 and n ≡ 1 (mod m) together imply that χ(n) = 1. If n ≡ 1 (mod m) then χ(n) = 1 (sends multiplicative identity to multiplicative identity), so m is a defining modulus, so the conductor of a Dirichlet character modulo m is less than or equal to m. We shall prove the Polya-Vinogradov inequality for primitive Dirchlet characters. The same inequality holds (using an O term rather than a particular constant) for non-primitive Dirichlet characters. The proof of that involves the fact [1, p. 152, Proposition 8] that a divisor m of m is a defining modulus for a Dirichlet character χ modulo m if and only if there exists a Dirichlet character χ modulo m such that χ(n) = χ 0 (n) · χ (n) n ∈ Z, where χ 0 is the principal Dirichlet character modulo m. (The principal Dirichlet character modulo m is that character such that χ(n) = 0 if gcd(n, m) > 1 and χ(n) = 1 otherwise.) If χ is a Dirichlet character modulo m, define the Gauss sum G(·, χ) : Z → C corresponding to this character by G(n, χ) = m−1 k=0 χ(k)e 2πikn/m , n ∈ Z. The Polya-Vinogradov inequality states that if χ is a primitive Dirichlet character modulo m, then n≤N χ(n) < √ m log m.