= Solution
Put $D_0=|D|$. The reference to part (c) in the printed hint is a reference to the positive-coefficient function from part (b). For $\sigma>1$ close to one, the preceding positivity and the supplied partial-fraction expansion give
$$
0\le\frac1{\sigma-1}+C\log D_0-\sum_{\substack{\beta\text{ real}\text{near }1}}\frac1{\sigma-\beta}.
$$
All omitted zero terms have nonnegative real parts because their real parts are at most one. The $O(1)$ zeta-pole remainder is included in $C\log D_0$, increasing the absolute constant if needed; a nonprincipal primitive real <conductor of a Dirichlet character> is at least three.
Suppose there were two real zeros, counted with multiplicity, with $\beta\ge1-c/\log D_0$. Set $\sigma=1+a/\log D_0$. Division by $\log D_0$ gives
$$
0\le\frac1a+C-\frac2{a+c}.
$$
Choose $C\ge1$, $a=1/(4C)$ and $c=a/4$. The right side is strictly negative. Thus
$$
\boxed{\text{at most one real zero, necessarily simple, in }[1-c/\log|D|,1].}
$$
This is the <uniqueness of a possible exceptional real Dirichlet zero>. It proves uniqueness, rather than existence of such a zero.
Back to article page
