\section{The $7/8$ hypothesis: proof of Theorem~\ref{thm:B}}\label{sec:qrh}

Throughout this section assume that no Dirichlet $L$-function has a zero with real part $> 7/8$. By the functional equation, every nontrivial
zero $\rho = \beta + i\gamma$ of a primitive $L(s, \chi)$ then satisfies $1/8 \le \beta \le 7/8$. The hypothesis is used only for
$L(s, \chi)$ with $\chi$ primitive modulo $q > 1$; no zero-free half-plane for $\zeta$ is needed.

\subsection{An explicit prime number theorem for characters}

\begin{proposition}\label{prop:psi}
Let $\chi$ be a primitive character modulo $q > 1$, $t \ge 16$ and $3 \le h \le t$, and put $T' = (t + h)/h$. Then
\begin{multline*}
 |\psi(t, \chi)| \le E(t, h, q) := \frac{2h \log(t+h)}{\log h} + \log(t+h)\Bigl(\frac{0.65 h}{\sqrt t} + \log\log_2(t+h) + 1\Bigr)
 + \frac{h}{2} + \frac32\\ + (t+h)^{7/8} Z_1(q, T') + 4.67\, Z_2(q) + \log(t+h) + 3.1,
\end{multline*}
where, with $a = 0.298$, $b = 4.358$, $c_0 = \log(2\pi e)$ and $N^+(1) = (\log q - c_0)/\pi + a \log q + b$,
\[
\begin{aligned}
 Z_1(q, T') &\le 7 N^+(1) + \tfrac{1}{\pi}\bigl[(\log q - c_0)\log T' + \tfrac12 \log^2 T'\bigr] + (a \log q + b)(1 - 1/T')\\ &\quad + a\bigl(1 - (\log T' + 1)/T'\bigr) + \tfrac{4}{\pi}\bigl(\log(q T') - c_0 + 1\bigr) + 2\bigl(a \log(q T') + a/2 + b\bigr)/T',\\
 Z_2(q) &\le 6.12\, N^+(1) + \tfrac{2}{\pi}(\log q - c_0 + 1) + a \log q + a/2 + b .
\end{aligned}
\]
\end{proposition}

\begin{proof}[Outline]
Smooth $\psi(t, \chi)$ by the ramp weight equal to $1$ on $[2, t]$ and linear on $[t, t+h]$; the difference is bounded by the
Brun--Titchmarsh inequality and the prime powers (the first terms). The explicit formula for the ramp sum runs over all zeros; the trivial zeros
give the last terms; for the nontrivial zeros the transform $W_t$ of the upper ramp satisfies
$|W_t(\rho)| \le (t+h)^{\beta}\min(1/|\rho|, 2T'/(|\rho||\rho+1|))$ with $\beta \le 7/8$ and $|\rho| \ge 1/8$, the fixed lower ramp
contributes at most $4.67/(|\rho||\rho+1|)$ per zero, hence $4.67\,Z_2(q)$ in all, and the zero counts of Bennett, Martin, O'Bryant and
Rechnitzer \cite{BMOR21} bound the resulting sums by $Z_1$ and $Z_2$.
The complete proof is in Appendix~\ref{app:psi}.
\end{proof}

\subsection{Distribution in progressions}

\begin{lemma}\label{lem:Q}
Let $N = e^L$ with $L \ge 300$, $\lambda > 0$, $H := \lambda N^{1/8}/L^3 < N$, and let $3 \le d \le H$ be odd squarefree with $(d, N) = 1$. Then
\[
 \Bigl| \pi(N; d, N) - \frac{\pi(N)}{\varphi(d)} \Bigr| \le \frac{c_E}{L}\Bigl(1 + \frac{32}{7L}\Bigr) N^{7/8} + 3\sqrt N + \frac{2\log H}{\log 3},
\]
where $c_E$ is any upper bound for $E(t, t^{7/8}, q)/t^{7/8}$ over $e^{L/2} \le t \le e^L$ and $1 < q \le H$.
\end{lemma}

\begin{proof}
By orthogonality, $\pi(N; d, N) - \pi(N)/\varphi(d) = -\omega(d)/\varphi(d) + \varphi(d)^{-1} \sum_{\chi \ne \chi_0} \bar\chi(N)\pi(N, \chi)$ with
$\pi(N, \chi) = \sum_{p \le N} \chi(p)$, and $|\pi(N, \chi) - \pi(N, \chi^*)| \le \omega(d)$ for the primitive $\chi^*$ inducing $\chi$;
here $\omega(d) \le \log H/\log 3$. By partial summation, $\pi(N, \chi^*) = \theta(N, \chi^*)/L + \int_2^N \theta(t, \chi^*)\,dt/(t\log^2 t)$
and $|\theta(t, \chi^*)| \le |\psi(t, \chi^*)| + \psi(t) - \theta(t)$. For $t \le \sqrt N$ use $|\theta(t, \chi^*)| \le \theta(t) < t\log 4$ (by (I3)),
contributing less than $(2/\log 2)\sqrt N < 2.885391\sqrt N$. For $t \ge \sqrt N$, Proposition~\ref{prop:psi} with $h = t^{7/8}$ gives
$|\psi(t, \chi^*)| \le c_E t^{7/8}$, and $\int_{\sqrt N}^N t^{-1/8}\,dt/\log^2 t \le (32/(7L^2)) N^{7/8}$. The prime powers contribute at most
$[\log 4\,(1/L + 8/L^2) + (2 + 12/L)e^{-L/6}]\sqrt N < 0.004745\sqrt N$ for $L \ge 300$, using $\psi(t) - \theta(t) = \sum_{k \ge 2}\theta(t^{1/k}) \le \log 4\,(\sqrt t + t^{1/3}\log_2 t)$,
again by (I3).
\end{proof}

\subsection{The sieve}
We use the explicit linear sieve of Bordignon, Johnston and Starichkova \cite[Theorem~6, Table~1]{BJS}, in the form
\cite[Theorem 2.1, Table 2.1]{Johnston} (whose Table 2.1 is their Table 1), in the setting of Johnston's Chapter 4: $\cA$ as above,
$\cP$ the primes not dividing $N$, $g(d) = 1/\varphi(d)$, $V(z) = \prod_{p < z,\, p \in \cP}(1 - g(p))$, and $Q = \{3, 5, 7, 11\} \cap \cP$.
By \cite[Proposition A.5]{Johnston} the hypothesis of his Theorem 2.1 holds with $\eps_J = 4.77 \cdot 10^{-3}$ for every $z \ge 694$ (for
$1 < u < 13$ the relevant product equals its value at $u = 13$), and $C_2(\eps_J) \le 154$ by Table 2.1. His error term is $\eps_J C_2(\eps_J) e^2 h(s)$ with $h(s) = e^{-s}$ on $[2, 3]$ and
$h(s) = 3s^{-1}e^{-s}$ for $s \ge 3$ \cite[(2.4)]{Johnston}, so $h(s) \le e^{-s}$ on $[2, 4]$. Thus for $D_s \ge z^2$ and
$s = \log D_s/\log z \in (2, 4]$,
\[
 S(\cA, z) > |\cA|\, V(z)\, \bigl(f(s) - \eps_J\, C_2(\eps_J)\, e^2 e^{-s}\bigr) - \sum_{\substack{d \mid P(z)\\ d < 1155 D_s}} |r(d)|,
 \qquad f(s) = \frac{2e^\gamma \log(s - 1)}{s},
\]
with $r(d) = |\cA_d| - |\cA|/\varphi(d)$. We also use $|\cA| > N/\log N$, $V(z) > \frac{U_N}{\log z}\bigl(1 - \frac{1.002 M}{z - 1}\bigr)
\bigl(1 - \frac{1}{2\log^2 z}\bigr)$ for $z = N^{1/M} \ge 285$, $\omega(N) \le 1.3841 \log N/\log\log N$, $\#\{d \le x \text{ odd squarefree}\}
\le 0.45 x$ ($x \ge 25$), and $|r(d)| \le |\pi(N; d, N) - \pi(N)/\varphi(d)| + \omega(N)$ \cite[Lemmas 4.1--4.4 and the proof of Lemma 4.16]{Johnston}.

\begin{lemma}\label{lem:T}
Let $N$ be even with $L = \log N \in [L_1, L_2]$, $L_1 \ge 300$, and fix integers $M \ge 2$ and $K \ge 1$, $0 < \lambda \le 100$ and $0 < \delta < 1$. Put
$z = N^{1/M}$, $H = \lambda N^{1/8}/L^3$, $D_s = H/1155$, $s = \log D_s/\log z$, and let $z_1, s_1$ be the values at $L_1$ and $s_2$ the value at
$L_2$. Suppose $H(L_1) \ge 25$ (then $H < N$, as $\lambda \le 100$ and $L \ge 300$), $z_1 \ge 10372$, $2 < s_1 \le s_2 \le 4$, $F(s_1) := f(s_1) - 4.77 \cdot 10^{-3} \cdot 154\, e^{2 - s_1} > 0$,
$1 - 1.002M/(z_1 - 1) > 0$, and
\[
 \mathrm{main} - \mathrm{rem} - e^{-L_1} - \mathrm{bad} > 0,
\]
where $\mathrm{main} = \frac{U_{\rm low} M}{L_2^2}\bigl(1 - \frac{1.002M}{z_1 - 1}\bigr)\bigl(1 - \frac{M^2}{2L_1^2}\bigr) F(s_1)$,
$\mathrm{rem} = \frac{0.45\lambda}{L_1^3}\bigl[\frac{c_E}{L_1}(1 + \frac{32}{7L_1}) + 3e^{-3L_1/8} + (\frac{2\log H(L_2)}{\log 3}
+ \frac{1.3841 L_2}{\log L_2}) e^{-7L_1/8}\bigr]$ with $c_E$ valid on the whole interval, and
\[
 \mathrm{bad} = e^{-\delta L_1} + \frac{1}{(1 - \delta) L_1} \sum_{k = K+1}^{M-1} \Bigl[ c_\theta \mu_{k-1} + \sum_{a=2}^{k} A_a(z_1)\, \mu_{k-a} \Bigr],
\]
with $\eps$ and the recursive $\mu_j$ of Lemma~\ref{lem:R3} taken at $z = z_1$. Then $N = p + \eta$ with $\eta \ge 2$, $P^-(\eta) \ge z$
and $\Omega(\eta) \le K$. If moreover $z_1 \ge 8 \cdot 10^9$, the same holds with $\mathrm{bad}$ replaced by
\[
 \mathrm{bad}_P = e^{-\delta L_1} + 4 e^{-L_1} + \frac{1}{(1 - \delta) L_1} \sum_{k = K+1}^{M-1} \Bigl[ G_P\Bigl(1 + \frac{480}{840 \log z_1}\Bigr) \mu_{k-1} + \sum_{a=2}^{k} A_a(z_1)\, \mu_{k-a} \Bigr],
 \qquad G_P = \frac{9}{32}.
\]
\end{lemma}

\begin{proof}
The sieve inequality, the lower bounds for $|\cA|$ and $V(z)$, and $h(s) \le e^{-s}$ on $[2, 4]$ give at least $(U_N/U_{\rm low})\,\mathrm{main}\cdot N$
for the main term; Lemma~\ref{lem:Q} and the counting bounds give at most $\mathrm{rem}\cdot N$ for the remainder, since $1155D_s \le H$ and the $d$
are odd squarefree. One sifted element may be $1$, and $1 \le e^{-L_1} N$. A $z$-rough $n < z^M$ has $\Omega(n) \le M - 1$. Those with
$\Omega(n) > K$ number at most $N^{1-\delta}$ with $n \le N^{1-\delta}$, plus, by Lemma~\ref{lem:R3}, at most $N \cdot (\mathrm{bad} - e^{-\delta L_1})$ others,
since every term of the bound decreases as $L$ grows: $1/L \le 1/L_1$, and $\eps$, $\mu_j$ and $A_a(z)$ decrease as $z$ grows. Every factor of
$\mathrm{main}$ and $\mathrm{rem}$ has the stated monotonicity on $[L_1, L_2]$: $s$ increases with $L$ for $\lambda \le 100$, $F$ increases on
$(2, 4]$ and $z$ increases. As $U_N \ge U_{\rm low}$, the first statement follows. For the second, Lemma~\ref{lem:P} bounds the elements with
$\Omega(n) > K$, apart from the at most four $n = N - q$, $q \in \{3, 5, 7, 11\}$, by $\Pi(N)$ times the corresponding expression; with
$U_N \ge U_{\rm low}\Pi(N)$ and $\Pi(N) \ge 1$, the good count exceeds the bad count plus one when $\mathrm{main} - \mathrm{rem} - e^{-L_1} - \mathrm{bad}_P > 0$
(Remark~\ref{rem:P}).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:B}]
For $N \le 4 \cdot 10^{18}$ use (I1); for $4 \cdot 10^{18} \le N \le e^{350}$ use Lemma~\ref{lem:C3} with $\delta = 1/5$ and $K = 14$
(Lemma~\ref{lem:C3num}). For $350 \le \log N \le 51046$ use Lemma~\ref{lem:T} with $K = 14$, $\delta = 0.05$ and $\mathrm{bad}$, on the $501$
intervals of \S\ref{sec:numerics}. For $51046 \le \log N \le 159011874857757$ (the right end exceeds $e^{32.7}$) use Lemma~\ref{lem:T} with
$K = 14$, $\delta = 0.05$ and $\mathrm{bad}_P$, on the $449$ intervals of \S\ref{sec:numerics}; there $z_1 \ge e^{51046/32} > 8 \cdot 10^9$. For
$N > \exp(\exp(32.7))$ use (I5).
\end{proof}

\begin{remark}
Without Lemma~\ref{lem:P} the top range gives $15$ but $14$ is not established for $\log N \ge 7.6 \cdot 10^{13}$; with it, $13$ is not
established on $103$ of the $449$ top intervals. Since the parameters are chosen by a search, these statements do not prove that $13$ or
$14$ fails.
\end{remark}