Pólya-Vinogradov bound is not tight for characters of odd, bounded order

You need to know: Set ${\mathbb Z}$ of integers, greatest common divisor (gcd) of 2 integers, set ${\mathbb C}$ of complex numbers, $o(1)$ notation.

Background: Function $\chi_q:{\mathbb Z}\to{\mathbb C}$ is called a Dirichlet character modulo integer $q>0$ if (i) $\chi_q(n)=\chi_q(n+q)$ for all n, (ii) if $\text{gcd}(n,q) > 1$ then $\chi_q(n)=0$ ; if $\text{gcd}(n,q) = 1$ then $\chi_q(n) \neq 0$ , and (iii) $\chi_q(mn)=\chi_q(m)\chi_q(n)$ for all integers m and n. An example is character $\chi_q^*$ such that $\chi_q^*(n)=1$ whenever $\text{gcd}(n,q) = 1$ and $\chi_q^*(n)=0$ otherwise. This character is called principal and all other characters – nonprincipal. The order of a Dirichlet character $\chi_q$ is the least positive integer m such that $\chi_q(n)^m=\chi_q^*(n)$ for all n. The character $\chi_q$ is called primitive if there is no integer $0<d<q$ such that $\chi_q(a)=\chi_q(b)$ whenever $\text{gcd}(a,q) = \text{gcd}(b,q) = 1$ and $a-b$ is a multiple of d. For any nonprincipal character $\chi_q$ let $M(\chi_q) = \max\limits_{1\leq m\leq q}\left|\sum\limits_{n=1}^m \chi_q(n)\right|$ .

The Theorem: On 2nd March 2005, Andrew Granville and Kannan Soundararajan submitted to the Jornal of the AMS a paper in which they proved that that if $\chi_q$ is a primitive character modulo q of odd order g, then $M(\chi_q) \leq C_g \sqrt{q} (\log q)^{1-\frac{\delta_g}{2} + o(1)}$ where $\delta_g = 1 - \frac{g}{\pi}\sin\frac{\pi}{g}$ , and $C_g$ is a constant depending only on g.

Short context: In 1918, Pólya and Vinogradov independently proved that $M(\chi_q)=O(\sqrt{q}\log q)$ for any nonprincipal Dirichlet character $\chi_q$ . This is known as the Pólya–Vinogradov inequality, has numerous applications, but it is an open question whether a better bound for $M(\chi_q)$ in terms of q is possible. The Theorem provides an improved bound for characters of odd, bounded order.