2 The setup

Our plan of attack goes along similar lines to those in [Reference Montgomery16, Ch. 12]. Let $\alpha>0$ be some fixed constant and $\mathcal {C}$ be a family of primitive Dirichlet characters, all with conductors not exceeding Q. Now define $\mathcal {R}$ to be a finite set of $(\,\rho ,\chi )$ such that $L(\,\rho ,\chi )=0$ for some $\chi \in \mathcal {C}$ , where $\beta \geq \sigma>\tfrac 12$ and $|\gamma |\leq T$ for all $(\,\rho ,\chi )\in \mathcal {R}$ and $|\gamma -\gamma '|\geq \alpha \log QT$ for some constant $\alpha $ and all distinct $(\,\rho ,\chi )$ and $(\,\rho ',\chi )\in \mathcal {R}$ .

Let $\{ a_n \}$ be a arbitrary sequence of complex numbers. We define

$$ \begin{align*} R(\,\chi)=\sum_{n\leq N}a_n\chi(n) \quad\text{and}\quad S(s,\chi)=\sum_{n\leq N} \frac{a_n \chi(n)}{n^s}. \end{align*} $$

If $\{ A_l \}$ and $\{ B_l \}$ are sequences of nonnegative real numbers and $L \in \mathbb {N}$ , then we set

$$ \begin{align*} \Delta (Q,N) = \sum_{l \leq L} Q^{A_l} N^{B_l} \quad \mbox{and} \quad \Delta_T (Q,N) = \sum_{l \leq L} Q^{A_l} N^{B_l} T^{1-B_l}. \end{align*} $$

With these conditions and notation, we can show, using the same arguments as those for (12.28) or (12.29) in [Reference Montgomery16], that it is possible to choose the elements of $\mathcal {R}$ so that, for any $\varepsilon>0$ ,

(2.1) $$ \begin{align} \sum_{\chi\in \mathcal{C}} N(\sigma,T,\chi) \ll (QT)^\varepsilon(|\mathcal{R}|+1), \end{align} $$

where the implied constant depends on $\varepsilon $ only. Consequently, our attention is shifted to estimating the size of $\mathcal {R}$ .

We define, for $X\geq 2$

$$ \begin{align*} M_X(s,\chi)=\sum_{n\leq X} \frac{\mu(n)\chi(n)}{n^s}, \end{align*} $$

where $\mu $ is the Möbius function. We note here that the Dirichlet series of $\mathfrak {M}_X(s,\chi )=L(s,\chi )M_X(s,\chi )$ has coefficients $\mathfrak {m}_{X,n}\chi (n)$ with

$$ \begin{align*} \mathfrak{m}_{X,n}=\sum_{\substack{d|n\\d\leq X}}\mu(d). \end{align*} $$

Thus $\mathfrak {m}_{X,1}=1$ , $\mathfrak {m}_{X,n}=0$ for $2\leq n\leq X$ , and $|\mathfrak {m}_{X,n}|\leq \tau (n)$ for $n>X$ , with $\tau $ denoting the divisor function.

Now we consider the Dirichlet series with coefficients $\mathfrak {m}_{X,n}\chi (n)e^{-n/Y}$ where $1\ll X\ll Y\ll (QT)^K$ for some sufficiently large $K\geq 1$ . From (12.25) and (12.26) of [Reference Montgomery16], for sufficiently large $\alpha =3A$ , each $(\,\rho ,\chi )\in \mathcal {R}$ satisfies at least one of the inequalities

(2.2) $$ \begin{align} \bigg|\sum_{X<n\leq Y^2}\mathfrak{m}_{X,n}\chi(n)n^{-\rho}e^{-n/Y}\bigg|\geq\frac{1}{6} \end{align} $$

and

(2.3) $$ \begin{align} \frac{1}{2\pi}\bigg|\int_{-A\log (QT)}^{A\log (QT)}\mathfrak{M}_X\bigg(\dfrac{1}{2}+i\gamma+iu,\chi\bigg)Y^{1/2-\beta+iu}\Gamma\bigg(\dfrac{1}{2}-\beta+iu\bigg) \,du\bigg|\geq\frac{1}{6}. \end{align} $$

Let $\mathcal {R}_1$ and $\mathcal {R}_2$ be the sets consisting of all elements of $\mathcal {R}$ satisfying (2.2) and (2.3), respectively. Hence,

(2.4) $$ \begin{align} | \mathcal{R} | \leq |\mathcal{R}_1| + |\mathcal{R}_2| \end{align} $$

and it suffices to estimate from above the sizes of $\mathcal {R}_1$ and $\mathcal {R}_2$ .

Along similar lines to the treatment in [Reference Montgomery16], we obtain

(2.5) $$ \begin{align} | \mathcal{R}_1 | \ll(\log Y)^3 \sum_{(\,\rho,\chi)\in\mathcal{R}_1}\bigg|\sum_{n=U}^{2U}\mathfrak{m}_{X,n}\chi(n)n^{-\rho}e^{-n/Y}\bigg|^2, \end{align} $$

for some U with $X \leq U\leq Y^2$ . For $\mathcal {R}_2$ , we get

(2.6) $$ \begin{align} | \mathcal{R}_2 | &\ll Y^{2/3-4\sigma/3} (QT)^\varepsilon\bigg(\sum_{(\,\rho,\chi)\in\mathcal{R}_2}\bigg|L\bigg(\dfrac{1}{2}+it_\rho,\chi\bigg)\bigg|^4\bigg)^{1/3} \bigg(\sum_{(\,\rho,\chi)\in\mathcal{R}_2} \bigg|M_X\bigg(\dfrac{1}{2}+it_\rho,\chi\bigg)\bigg|^2\bigg)^{2/3}. \end{align} $$

Here, for each $(\,\rho ,\chi ) \in \mathcal {R}_2$ , $t_{\,\rho }$ is defined to be the real number in the interval $[\gamma - A \log (QT), \gamma +A \log (QT)]$ for which $|\mathfrak {M}_X(\tfrac 12+it_{\,\rho },\chi )|$ is maximum.

Now we are led to estimate sums of the form

(2.7) $$ \begin{align} \sum_{\chi\in \mathcal{C}} \sum_{s\in\mathcal{S}_\chi}\bigg|\sum_{n\leq N} \frac{a_n\chi(n)}{n^s}\bigg|^2, \end{align} $$

where $\mathcal {S}_\chi $ is a set of complex numbers. To that end, various kinds of large sieve inequalities will play an indispensable role. We refer the reader to [Reference Montgomery16, Reference Ramaré18] and [Reference Iwaniec and Kowalski14, Ch. 7] for more extensive discussions on the large sieve, a subject of independent interest.

We first write down a general result for sums of the form (2.7).

Lemma 2.1. Let $\mathcal {C}$ be an arbitrary set of primitive Dirichlet characters with conductors at most Q, and $\mathcal {S}_\chi $ be a finite set of complex numbers $s=\sigma +it$ . Suppose $T_0$ , T, $\sigma _0>\delta >0$ are such that $T_0+\delta /2 \leq |t|\leq T_0+T-\delta /2$ for all $s\in \mathcal {S}_\chi $ , $1/2 \leq \sigma _0 \leq \sigma \leq ~1$ for all $s\in \mathcal {S}_\chi $ and $|t-t'|\geq \delta $ for distinct s, $s'\in \mathcal {S}_\chi $ . If the bound

$$ \begin{align*} \sum_{\chi \in \mathcal{C}} | R(\,\chi)|^2 \ll \Delta (Q,N) \sum_{n \leq N} |a_n|^2 \end{align*} $$

holds, then we have

$$ \begin{align*} \sum_{\chi \in \mathcal{C}} \sum_{s \in \mathcal{S}_\chi} | S(s, \chi)|^2 \ll \bigg( \frac{1}{\delta} + \log N \bigg) \Delta_T(Q,N) \sum_{n \leq N} \frac{|a_n|^2}{ n^{2\sigma_0}} \bigg(1+\log\frac{\log2N}{\log2n}\bigg). \end{align*} $$

Proof. The proof is rather standard and thus we only give a sketch here. Let

$$ \begin{align*} S_u(s,\chi) = \sum_{2 \leq n \leq u} a_n \chi(n) n^{-s}. \end{align*} $$

Partial summation and Cauchy’s inequality give

$$ \begin{align*} |S_u(s,\chi)|^2 \ll |a_1|^2 + |S(\sigma_0+it, \chi)|^2 + \int\nolimits_2^N |S_u(\sigma_0+it , \chi)|^2 \frac{\,du}{u \log u}. \end{align*} $$

Using [Reference Montgomery16, Lemma 1.4], we get

(2.8) $$ \begin{align} \sum_{\chi \in \mathcal{C}} \sum_{t \in \mathcal{T}_\chi} | S(it, \chi)|^2 \ll \bigg( \frac{1}{\delta} + \log N \bigg) \sum_{\chi \in \mathcal{C}} \int_{T_0}^{T_0+T} |S(it, \chi)|^2 \,dt \end{align} $$

where $\mathcal {T}_{\chi } = \{ t : s = \sigma +it \in \mathcal {S}_{\chi } \}$ . Now arguing along similar lines to the proof of [Reference Gallagher7, Theorem 2], we arrive at

(2.9) $$ \begin{align} \sum_{\chi \in \mathcal{C}} \int_{T_0}^{T_0+T} |S(it, \chi)|^2 \,dt \ll \Delta_T(Q,N) \sum_{n \leq N} |a_n|^2. \end{align} $$

The desired bound follows easily by combining all the bounds above.

From this discussion, we have the following general result which can be used to derive a zero density result for any collection of primitive Dirichlet characters if the corresponding large sieve inequality and bound for the fourth moment of L-functions are available.

Theorem 2.2. Let $\mathcal {C}$ be a finite family of primitive Dirichlet characters, none of which have conductors greater than Q, and suppose that

$$ \begin{align*} \sum_{\chi\in\mathcal{C}}|R(\,\chi)|^2\ll\Delta(Q,N)\sum_{n\leq N}|a_n|^2\quad\text{and}\quad\sum_{\chi\in\mathcal{C}}\int_{-T}^{T}\bigg|L\bigg(\dfrac{1}{2}+it,\chi\bigg)\bigg|^4\,dt \ll\mathfrak{L} \end{align*} $$

hold. Then, for any $\sigma $ with $\tfrac {1}{2} \leq \sigma \leq 1$ , and $X,Y$ satisfying $1\ll X\ll Y\ll (QT)^K$ for some absolute constant K, there is a U with $X\ll U\ll Y^2$ such that

$$ \begin{align*} \sum_{\chi\in\mathcal{C}}N(\sigma,T,\chi)\ll (QT)^\varepsilon(\mathfrak{L}^{1/3} \Delta_T(Q,X)^{2/3}Y^{2(1-2\sigma)/3}+\Delta_T(Q,U)U^{1-2\sigma}e^{-2U/Y}), \end{align*} $$

where the implied constant depends on $\varepsilon $ alone.

Proof. We take $\delta =3A\log QT$ in Lemma 2.1, where A is as in (2.3), and obtain

(2.10) $$ \begin{align} \sum_{(\,\rho,\chi)\in\mathcal{R}_1}\bigg|\sum_{n=U}^{2U}\mathfrak{m}_{X,n}\chi(n)n^{-\rho}e^{-n/Y}\bigg|^2\ll(QT)^\varepsilon\Delta_T(Q,U)U^{1-2\sigma}e^{-2U/Y} \end{align} $$

and

(2.11) $$ \begin{align} \sum_{(\kern1.1pt\rho,\chi)\in\mathcal{R}_2}\bigg|M_X\bigg(\dfrac{1}{2}+it,\chi\bigg)\bigg|^2\ll(QT)^\varepsilon\Delta_T(Q,X). \end{align} $$

Using similar methods to [Reference Montgomery16, Theorem 10.3], we can show that

(2.12) $$ \begin{align} \sum_{(\kern1.1pt\rho,\chi)\in\mathcal{R}_2}\bigg|L\bigg(\dfrac{1}{2}+it,\chi\bigg)\bigg|^4\ll(QT)^\varepsilon\mathfrak{L}. \end{align} $$

Now, from (2.5) and (2.10) we obtain a bound for $|\mathcal {R}_1|$ , and (2.6), (2.11) and (2.12) give rise to a majorant for $|\mathcal {R}_2|$ . The result now follows from (2.1) and (2.4).

Our second general result below does not require any large sieve-type bound.

Theorem 2.3. Let $\mathcal {C},Q,T,\mathfrak {L}$ be as in Theorem 2.2. Then, for any $\sigma $ with $\tfrac {1}{2}<\sigma \leq 1$ and any $\varepsilon>0$ ,

$$ \begin{align*} \sum_{\chi\in\mathcal{C}}N(\sigma,T,\chi)\ll(QT)^\varepsilon((\mathfrak{L}Q^2T)^{(1-\sigma)/\sigma}+(Q^2T)^{(1-\sigma)/(2\sigma-1)}), \end{align*} $$

where the implied constant depends on $\varepsilon $ alone.

3 Proof of Theorems 1.1 and 1.3

Before proving Theorems 1.1 and 1.3, we need the following lemma.

Lemma 3.1. Let $\mathcal {Q}$ be as above. Then, for any $T\geqslant 2$ and $\varepsilon>0$ ,

$$ \begin{align*} \sum_{q\in\mathcal{Q}} \ \sideset{}{^*}\sum_{\chi\bmod{q}}\int_{-T}^{T}\bigg|L\bigg(\dfrac{1}{2}+it,\chi\bigg)\bigg|^4\,dt\ll|\mathcal{Q}|(QT)^{1+\varepsilon}, \end{align*} $$

where the implied constant depends on $\varepsilon $ alone.

We now proceed with the proof of Theorem 1.1.

Proof of Theorem 1.1

From [Reference Baier1, Theorem 2] we have

(3.1) $$ \begin{align} \sum_{q\in\mathcal{Q}} \ \sideset{}{^*}\sum_{\chi \bmod{q}}|R(\,\chi)|^2\ll(QN)^\varepsilon(|\mathcal{Q}|Q+N+QN^{1/2})\sum_{n\leq N}|a_n|^2. \end{align} $$

Moreover, the classical large sieve inequality gives (see the discussion around (5.4) and (5.5) in [Reference Zhao19]),

(3.2) $$ \begin{align} \sum_{q\in\mathcal{Q}} \ \sideset{}{^*}\sum_{\chi \bmod{q}}|R(\,\chi)|^2\ll \min(|\mathcal{Q}|(Q+N),Q^2+N)\sum_{n\leq N}|a_n|^2. \end{align} $$

Using (3.1), Lemma 3.1 and Theorem 2.2, we obtain

$$ \begin{align*} \sum_{q\in\mathcal{Q}} \ \sideset{}{^*}\sum_{\chi\bmod{q}}N(\sigma,T,\chi) & \ll(QT)^\varepsilon\bigg((|\mathcal{Q}|QT)^{1/3}(|\mathcal{Q}|QT+X+QT^{1/2}X^{1/2})^{2/3}Y^{2(1-2\sigma)/3} \\[-5pt] &\quad +|\mathcal{Q}|QTX^{1-2\sigma}+Y^{2-2\sigma} +QT^{1/2}\begin{cases}Y^{3/2-2\sigma}&\text{if }\tfrac{1}{2}\leq\sigma\leq\tfrac{3}{4}\\X^{3/2-2\sigma}&\text{otherwise}\end{cases}\bigg). \end{align*} $$

On taking

$$ \begin{align*} X=|\mathcal{Q}|^2T\quad\text{and}\quad Y=|\mathcal{Q}|^{6/(5-4\sigma)}T^{3/2(2-\sigma)} \end{align*} $$

in the case $1/2 \leq \sigma \leq 3/4$ , and

$$ \begin{align*} X=|\mathcal{Q}|^{(2\sigma-2)/(9\sigma-4(\sigma^2+1))}Q^{(4\sigma-2)/(9\sigma-4(\sigma^2+1))}T \end{align*} $$

and

$$ \begin{align*} Y=|\mathcal{Q}|^{(4\sigma-3)/(18\sigma-8(\sigma^2+1))}Q^{(12\sigma-7)/(18\sigma-8(\sigma^2+1))}T^{3/2(2-\sigma)} \end{align*} $$

in the other case, we obtain

(3.3) $$ \begin{align} \sum_{q\in\mathcal{Q}} \ \sideset{}{^*}\sum_{\chi\bmod{q}}N(\sigma,T,\chi)\ll(QT)^\varepsilon\begin{cases}|\mathcal{Q}|^{3(3-4\sigma)/(5-4\sigma)}QT^{3(1-\sigma)/(2-\sigma)}, \\ (|\mathcal{Q}|^{4\sigma-3}Q^{12\sigma-7})^{(1-\sigma)/(9\sigma-4(\sigma^2+1))}T^{3(1-\sigma)/(2-\sigma)}, \end{cases} \end{align} $$

respectively, in the case $1/2 \leq \sigma \leq 3/4$ and in the other case. Now, using (3.2) and Lemma 3.1 in Theorem 2.2, we also have

$$ \begin{align*} \sum_{q\in\mathcal{Q}} \ \sideset{}{^*}\sum_{\chi\bmod{q}}N(\sigma,T,\chi) &\ll(QT)^\varepsilon( (|\mathcal{Q}|QT)^{1/3}\min(|\mathcal{Q}|QT+|\mathcal{Q}|X,Q^2T+X)^{2/3}Y^{2(1-2\sigma)/3} \\[-5pt] &\quad +\min(|\mathcal{Q}|QTX^{1-2\sigma}+|\mathcal{Q}|Y^{2-2\sigma},Q^2TX^{1-2\sigma}+Y^{2-2\sigma})). \end{align*} $$

We first take

$$ \begin{align*} X=QT\quad\text{and}\quad Y=X^{3/2(2-\sigma)} \end{align*} $$

and then

$$ \begin{align*} X=Q^2T\quad\text{and}\quad Y=X^{3/2(2-\sigma)}, \end{align*} $$

and on comparing these results, we arrive at

(3.4) $$ \begin{align} \sum_{q\in\mathcal{Q}} \ \sideset{}{^*}\sum_{\chi\bmod{q}}N(\sigma,T,\chi)\ll(QT)^\varepsilon\min(|\mathcal{Q}|Q^{3(1-\sigma)/(2-\sigma)},Q^{6(1-\sigma)/(2-\sigma)})T^{3(1-\sigma)/(2-\sigma)}. \end{align} $$

Our desired result follows from comparing (3.3) with (3.4) and Theorem 2.3.

Now, considering the case in which $\mathcal {Q}$ is the set of k-power moduli, from [Reference Baier and Zhao3, Theorem 1], we have, for any integer $k\geq 3$ and any Q, $\varepsilon>0$ ,

(3.5) $$ \begin{align} \sum_{q\leq Q} \ \sideset{}{^*}\sum_{\chi \bmod{q^k}}|R(\,\chi)|^2\ll(QN)^\varepsilon(Q^{k+1}+N+Q^kN^{1/2})\sum_{n\leq N}|a_n|^2, \end{align} $$

which is just a special case of (3.1). Thus (3.5) will lead to a result already contained in Theorem 1.1 and gives precisely Corollary 1.2. We note here that (3.5) has been improved in certain ranges by a number of authors: Halupczok [Reference Halupczok10, Reference Halupczok11], Munsch [Reference Munsch17], Halupczok and Munsch [Reference Halupczok and Munsch12], and Baker et al. [Reference Baker, Munsch and Shparlinski5]. Unfortunately, using the method here, the results in [Reference Baker, Munsch and Shparlinski5, Reference Halupczok10, Reference Halupczok11, Reference Halupczok and Munsch12] do not lead to any outcome better than Corollary 1.2.

In the case of square moduli, the best available large sieve inequality is found in [Reference Baier and Zhao4],

$$ \begin{align*} \sum_{q\leq Q}\ \sideset{}{^*}\sum_{\chi\bmod{q^2}}|R(\,\chi)|^2\ll(QN)^\varepsilon(Q^3+N+\min(N\sqrt{Q},Q^2\sqrt{N}))\sum_{n\leq N}|a_n|^2, \end{align*} $$

and from this we can derive Theorem 1.3, which is better than what Theorem 1.1 gives in certain regions.

Proof of Theorem 1.3

By Theorem 2.2,

$$ \begin{align*} \begin{split} &\sum_{q\leq Q} \ \sideset{}{^*}\sum_{\chi\bmod{q^2}}N(\sigma,T,\chi) \\ & \ll(QT)^\varepsilon \bigg( (Q^3T)^{1/3}(Q^3T+X+\min(Q^{1/2}X,Q^2T^{1/2}X^{1/2}))^{2/3}Y^{2(1-2\sigma)/3}+Q^3TX^{1-2\sigma} \\ & \hspace{3.7cm} +Y^{2-2\sigma}+\min\bigg(Q^{1/2}Y^{2-2\sigma},Q^2\begin{cases}Y^{3/2-2\sigma}&\text{if }\sigma\leq\tfrac{3}{4}\\X^{3/2-2\sigma}&\text{otherwise}\end{cases}\bigg) \bigg). \end{split} \end{align*} $$

To get the desired result, we simply take

$$ \begin{align*} X=Q^2T\quad\text{and}\quad Y=Q^{15/4(2-\sigma)}T^{3/2(2-\sigma)} \end{align*} $$

if $\sigma \leq \tfrac {3}{4}$ , and

$$ \begin{align*} X=Q^{(10\sigma-6)/(9\sigma-4(\sigma^2+1))}T\quad\text{and}\quad Y=Q^{(28\sigma-17)/(18\sigma-8(\sigma^2+1))}T^{3/2(2-\sigma)} \end{align*} $$

in the latter case. Hence,

(3.6) $$ \begin{align} \sum_{q\leq Q} \ \sideset{}{^*}\sum_{\chi\bmod{q^2}}N(\sigma,T,\chi)\ll(QT)^\varepsilon \begin{cases}Q^{(17-16\sigma)/2(2-\sigma)}T^{3(1-\sigma)/(2-\sigma)}& \text{if }\tfrac{1}{2}\leq\sigma\leq\tfrac{3}{4}\\ Q^{(1-\sigma)(28\sigma-17)/(9\sigma-4(\sigma^2+1))}T^{3(1-\sigma)/(2-\sigma)}& \text{otherwise.} \end{cases} \end{align} $$

The result follows on comparing (3.6) with Theorem 1.1.

In [Reference Zhao19], an optimal conjectural large sieve inequality for power moduli is given and yields the bound

(3.7) $$ \begin{align} \sum_{q\leq Q} \ \sideset{}{^*}\sum_{\chi \bmod{q^k}}|R(\,\chi)|^2\ll Q^\varepsilon(Q^{k+1}+N)\sum_{n\leq N}|a_n|^2. \end{align} $$

If (3.7) holds, then

$$ \begin{align*} \sum_{q\leq Q} \ \sideset{}{^*}\sum_{\chi \bmod{q^k}} N(\sigma,T,\chi)\ll(Q^{k+1}T)^{3(1-\sigma)/(2-\sigma)+\varepsilon} \end{align*} $$

holds for all positive Q and T.

4 Proof of Theorem 1.4

To establish Theorem 1.4, we require, in view of (2.5) and (2.6), bounds for

$$ \begin{align*} \sum_{\chi \in \mathcal{C}} | R(\,\chi) |^2 \quad \mbox{and} \quad \sum_{\chi \in \mathcal{C}} \bigg| L \bigg( \dfrac{1}{2} + it , \chi\bigg) \bigg|^4 , \end{align*} $$

where $\mathcal {C}$ is the family of characters under consideration.

For $\mathcal {C}_2 (Q)$ , using [Reference Heath-Brown13, Corollary 3], we get

(4.1) $$ \begin{align} \sum_{\chi \in \mathcal{C}_2(Q)} | R(\,\chi) |^2 \ll (QN)^{\varepsilon} (QN +N^2) \max_{n \leq N} |a_n|^2. \end{align} $$

By setting $\sigma =1/2$ in Theorem 2 of [Reference Heath-Brown13], for $T>1$ and $|t| \leq T$ ,

(4.2) $$ \begin{align} \sum_{\chi \in \mathcal{C}_2(Q)} \bigg| L \bigg( \dfrac{1}{2} + it , \chi\bigg) \bigg|^4 \ll (QT)^{1+\varepsilon}. \end{align} $$

Using (4.1) and Lemma 2.1 with some minor changes (the bound (4.1) is formally different from what is in the condition of Lemma 2.1), we get

(4.3) $$ \begin{align} \sum_{(\kern1.2pt\rho,\chi)\in\mathcal{R}_1}\bigg|\sum_{n=U}^{2U}\mathfrak{m}_{X,n}\chi(n)n^{-\rho}e^{-n/Y}\bigg|^2 \ll (QT)^{\varepsilon} (QTX^{1-2\sigma} +Y^{2-2\sigma} ). \end{align} $$

Now (2.8) and (2.9), together with [Reference Heath-Brown13, Corollary 1], produce the bound

(4.4) $$ \begin{align} \sum_{(\kern1.1pt\rho,\chi)\in\mathcal{R}_2}\bigg|M_X\bigg(\dfrac{1}{2}+it_\rho,\chi\bigg)\bigg|^2 \ll (QT)^{\varepsilon} (QT+X). \end{align} $$

Substituting (4.2), (4.3) and (4.4) into (2.5) and (2.6), we get

$$ \begin{align*} \sum_{\chi\in \mathcal{C}_2(Q)} N(\sigma, T, \chi) \ll (QT)^{\varepsilon} ( (QT^2)^{1/3} (QT+X)^{2/3} Y^{2(1-2\sigma)/3} + QT X^{1-2\sigma} +Y^{2-2\sigma}). \end{align*} $$

Setting

$$ \begin{align*} X=QT \quad \mbox{and} \quad Y= (Q^3T^4)^{1/(2(2-\sigma))} , \end{align*} $$

we arrive at the first term in the minimum in (1.2).

In the case of $\mathcal {C}_j (Q)$ with $j=3$ , $4$ and $6$ , we use the results in [Reference Baier and Young2, Reference Gao and Zhao9]. The only minor obstruction is that one requires the sum over n in $R(\,\chi )$ to be over square-free n. This is easily handled by rewriting $n=kl^2$ with k square-free and applying Cauchy’s inequality and then utilising the large sieve inequalities for cubic, quartic and sextic characters. For $j=3$ and $6$ , from Theorems 1.4 and 1.5 of [Reference Baier and Young2],

(4.5) $$ \begin{align} \sum_{\chi \in \mathcal{C}_j(Q)} | R(\,\chi) |^2 \ll (QN)^{\varepsilon} \min \{ Q^{5/2} N^{1/2} +N^{3/2}, Q^{11/9} + Q^{2/3} N \} \sum_{n \leq N} |a_n|^2. \end{align} $$

With $j=4$ , from Lemma 2.10 of [Reference Gao and Zhao9], which is an improvement of [Reference Gao and Zhao8, Theorem 1.2], we arrive at the bound

(4.6) $$ \begin{align} \sum_{\chi \in \mathcal{C}_4(Q)} | R(\,\chi) |^2 \ll (QN)^{\varepsilon} \min\{ Q^{3/2} N^{1/2} +N^{3/2}, Q^{7/6} + Q^{2/3} N \} \sum_{n \leq N} |a_n|^2. \end{align} $$

The results in [Reference Baier and Young2, Reference Gao and Zhao9] have more terms in the minimum than those given in (4.5) and (4.6). Here, we only cite what we will use later.

If $j=3$ or $6$ , then for all $T\gg Q^{2/3}$ ,

(4.7) $$ \begin{align} \sum_{\chi \in \mathcal{C}_j(Q)} \bigg| L \bigg( \dfrac{1}{2} + it , \chi\bigg) \bigg|^4 \ll (QT)^{3/2+\varepsilon}. \end{align} $$

The bound (4.7) also holds if $j=4$ and $T\gg Q^{1/2}$ . The proof of (4.7) uses the same arguments as in [Reference Heath-Brown13, Theorem 2]. The only difference is that, instead of (4.1), one uses (4.5) or (4.6) with the first terms in the minima at the appropriate places.

Now proceeding in the same way as for $\mathcal {C}_2(Q)$ , using (4.7) and the second terms in the minima given in the bounds (4.5) and (4.6), we deduce

(4.8) $$ \begin{align}\begin{split} \sum_{\chi \in \mathcal{C}_j(Q)} N(\sigma, T, \chi) & \ll (QT)^{\varepsilon} ( (Q^{3/2}T^{5/2})^{1/3} (Q^{11/9} T + Q^{2/3}X)^{2/3} Y^{2(1-2\sigma)/3} \\ &\quad + Q^{11/9} T \max (X^{3/2-2\sigma}, Y^{3/2-2\sigma} ) + Q^{2/3}Y^{5/2-2\sigma} ) \end{split}\end{align} $$

for $j = 3$ , $6$ , and

(4.9) $$ \begin{align} \begin{split}\sum_{\chi \in \mathcal{C}_4(Q)} N(\sigma, T, \chi) & \ll (QT)^{\varepsilon} ( (Q^{3/2}T^{5/2})^{1/3} (Q^{7/6} T + Q^{2/3}X)^{2/3} Y^{2(1-2\sigma)/3} \\ & \quad + Q^{7/6} T \max (X^{3/2-2\sigma}, Y^{3/2-2\sigma} ) + Q^{2/3}Y^{5/2-2\sigma}). \end{split}\end{align} $$

Taking

$$ \begin{align*} X = Q^{5/27}{T^{1/2}} \quad \mbox{and} \quad Y = Q^{5/(45-36\sigma)} T^{12/(11-4\sigma)} \end{align*} $$

in (4.8) and

$$ \begin{align*} X = Q^{2/9}{T^{1/2}} \quad \mbox{and} \quad Y = Q^{2/(15-12\sigma)} T^{12/(11-4\sigma)} \end{align*} $$

in (4.9), we get the first terms in the minima in (1.3) and (1.4). The second terms in the minima in (1.2), (1.3) and (1.4) are derived from Theorem 2.3 and either (4.2) or (4.7). This concludes the proof of Theorem 1.4.

Jutila [Reference Jutila15, Theorem 2] previously gave the bound

(4.10) $$ \begin{align} \sum_{\chi\in\mathcal{C}_2(Q)}N(\sigma,T,\chi)\ll(QT)^{(7-6\sigma)/(6-4\sigma)+\varepsilon} \end{align} $$

without the advantage of the mean value estimate (4.1). After proving (4.1), Heath-Brown [Reference Heath-Brown13, Theorem 3] was able to improve the Q-aspect of (4.10) to

(4.11) $$ \begin{align} \sum_{\chi\in\mathcal{C}_2(Q)}N(\sigma,T,\chi)\ll(QT)^\varepsilon Q^{3(1-\sigma)/(2-\sigma)}T^{(3-2\sigma)/(2-\sigma)}. \end{align} $$

However, (4.11) was obtained by first bounding the number of zeros in the subregions

$$ \begin{align*}\{\kern1.1pt\rho:\sigma\leq\beta<\sigma+(\log QT)^{-1},\tau\leq\gamma<\tau+(\log QT)^{-1}\}\quad\text{with}\ |\tau|\leq T\end{align*} $$

and then summing trivially over these subregions to obtain a bound for the total number of zeros in the rectangle $\{\kern1.1pt\rho :\sigma \leq \beta \leq 1,|\gamma |\leq T\}$ . By considering the whole rectangle from the start and employing Lemma 2.1 to average over the $\rho $ in the rectangle, we are able to improve the T-aspect of (4.11) in our result (1.2). Moreover, (1.2) is an improvement of (4.10) when $Q^{-4+11\sigma -6\sigma ^2}\gg T^{-10+21\sigma -10\sigma ^2}$ , which is true for all Q, $T>1$ when $\sigma \geq {(21-\sqrt {41})}/{20} \approx 0.7298$ .

We end the paper with the following remark. Recent heuristics in [Reference Dunn and Radziwiłł6] gave rise to some surprising revelations on the true optimal bound in the large sieve inequality for cubic Hecke characters, based on which, as well as its quartic analogue, the estimates in (4.5) and (4.6) are derived. Thus it gives one pause in conjecturing what the best possible form of the large sieve inequality for cubic and quartic Dirichlet characters should be. Consequently, unlike Theorem 1.1, it is unclear what the best possible unconditional bounds one can hope for in (1.3) and (1.4) may be using the methods of this paper.