Let $\chi\_{q}$ be a primitive Dirichlet character with modulus $q$ [(see definition at wikipedia )](https://en.wikipedia.org/wiki/Dirichlet_character).
For example for $q=5$ we have


\begin{equation}
\begin{aligned}
\chi\_{5,1}&=(1, 1, 1, 1, 0),\\
\chi\_{5,2}&=(1, i, -i, -1, 0),\qquad\qquad \text{(1)}\\
\chi\_{5,3}&=(1, -1, -1, 1, 0),\\
\chi\_{5,4}&=(1, -i, i, -1, 0),\\
\end{aligned}
\end{equation}


We construct a sum involving Dirichlet character $\chi\_{q}$ as


\begin{equation}
\begin{aligned}
Q(\chi\_{q})&=\sum\_{k=1}^{q-1}k\chi\_q(k)\qquad\qquad \text{(2)}
\end{aligned}
\end{equation}


**Proposition A:**
For a complex $\chi\_q$, like $\chi\_{5,2},\chi\_{5,4}$,
\begin{equation}
\begin{aligned}
&\color{red}{\text{sign}(\mathrm{Re}\chi\_{q}(-1))}Q(\mathrm{Re}\chi\_{q})>0,\qquad\text{(3)}\\
&\color{red}{\text{sign}(\mathrm{Im}\chi\_{q}(-2))}Q(\mathrm{Im}\chi\_{q})>0.\qquad\text{(4)}\\
\end{aligned}
\end{equation}


We are seeking a proof or a reference of the proof for this proposition.


This problem is related to a partial answer of another problem titled [sign unchanged for Dirichlet polynomials?](https://math.stackexchange.com/questions/3297969/sign-unchanged-for-dirichlet-polynomials) that we posted earlier at math.stackexchange.com. The motivation of studying this problem is also mentioned [there](https://math.stackexchange.com/questions/3297969/sign-unchanged-for-dirichlet-polynomials).



The sign of an interesting sum involving a Dirichlet character
(3) and (4) are false in general, even if we weaken $>$ to $\geq$. Let $\zeta:=e^{i\pi/8}$ be a primitive $16$-th root of unity, and let $\chi$ be the unique primitive Dirichlet character modulo $17$ satisfying $\chi(3)=\zeta^5$. Then $\chi(-1)=-1$, and
$$(\chi(1),\chi(2),\chi(3),\chi(4),\chi(5),\chi(6),\chi(7),\chi(8))=(1,\zeta^6,\zeta^5,\zeta^{12},\zeta^9,\zeta^{11},\zeta^7,\zeta^2).$$
However,
\begin{align\*}Q(\chi)&=\sum\_{k=1}^{16} k\chi(k)=\sum\_{k=1}^{8} (2k-17)\chi(k)\\[6pt]
&=-(15+13\zeta^6+11\zeta^5+9\zeta^{12}+7\zeta^9+5\zeta^{11}+3\zeta^7+\zeta^2)\\[10pt]
&\approx \ 8.84701161719 - 4.91203840222\,i.
\end{align\*}
So $Q(\chi)$ has positive real part, even though $\chi(-1)=-1$. This contradicts (3). Similarly, if we change $\zeta^5$ to $\zeta^3$ in the definition of $\chi$, we get a counterexample to (4).


