3.1 Lower bounds for weighted GCD sums

We extend the construction introduced by de la Bretèche and Tenenbaum [Reference de la Bretèche and Tenenbaum12]. Define

(3-1) $$ \begin{align} S_{1/2}(\mathcal{M},a_{\mathbb{K}}) {:=}q \sum_{m,n\in \mathcal{M}} a_{\mathbb{K}}\bigg(\frac{m}{(m,n)}\bigg)a_{\mathbb{K}}\bigg(\frac{n}{(m,n)}\bigg)\sqrt{\frac{(m,n)}{[m,n]}}. \end{align} $$

Let $\alpha \in (1,e)$ and $\delta \in (0,1)$ be some parameters, and let $N = \lfloor T^{\lambda } \rfloor $ be an integer not larger than T with $\lambda $ to be chosen later. Let k be an integer satisfying $1\le k \le (\log _{2}N)^{\delta },$ then define $P_k$ as the set of primes p that satisfy

$$ \begin{align*} \phi(d)\alpha^{k}\log N \log_{2}N < p \le \phi(d)\alpha^{k + 1}\log N \log_{2}N \quad \text{and} \quad p\equiv 1 \quad\pmod{\ d}. \end{align*} $$

Since $d \ll (\log _2 T)^A $ , the Siegel–Walfisz theorem implies

$$ \begin{align*} |P_{k}| \sim \alpha^{k}(\alpha - 1)\log N \bigg(1 + O\bigg(\frac{k+\log_{3}N}{\log_{2}N}\bigg)\bigg). \end{align*} $$

Let $\beta> 1$ satisfy $\beta \delta \log \alpha < 1$ . We then define

$$ \begin{align*} W_k {:=}q 2\bigg\lfloor\frac{\beta\log N}{2k^{2}\log_{3}N}\bigg\rfloor. \end{align*} $$

Furthermore, we let $W {:=}q \prod _{p\in P_{k}}p$ and consider the following set:

$$ \begin{align*} \mathcal{M}_k{:=}q \bigg\{m:m = \frac{\ell}{q}W, \ \omega(\ell)\le \frac{1}{2}W_{k}, \omega(q) \le \frac{1}{2}W_{k}, \ \ell q\mid W\bigg\}, \end{align*} $$

where $\omega (n)$ denotes the number of distinct prime factors of n. Next, we consider

(3-2) $$ \begin{align} \mathcal{M}{:=}q\bigg\{m = \prod_{1\leq k\leq (\log_{2}N)^{\delta}}m_{k}: m_{k}\in\mathcal{M}_k ,\ 1\leq k\leq (\log_{2}N)^{\delta}\bigg\}. \end{align} $$

According to [Reference de la Bretèche and Tenenbaum12, Lemma 2.2], we have $|\mathcal {M}|\le N$ . We establish the following lower bound for $S_{1/2}(\mathcal {M},a_{\mathbb {K}})$ .

Proposition 3.1. Define $S_{1/2}(\mathcal {M},a_{\mathbb {K}})$ as in (3-1), where $\mathcal {M}$ is given in (3-2). Then, for $d \ll (\log _2 T)^A$ ,

$$ \begin{align*}\frac{S_{1/2}(\mathcal{M}, a_{\mathbb{K}})}{|\mathcal{M}|} \ge \exp\bigg(\bigg(2\sqrt{\phi(d)} + o(1)\bigg)\sqrt{\frac{\log N \log_3 N}{\log_2 N}} \bigg)\end{align*} $$

as $N \to \infty $ .

Fonga attempted to extend the results of de la Bretèche and Tenenbaum in [Reference Fonga17, Theorem 1.1]. We argue that the range of k he chose is inappropriate. Even so, the proof of this proposition still follows the idea of Fonga; see [Reference Fonga17, pp. 3–7]. Specifically, note that $a_{\mathbb {K}}(n)$ is not a completely multiplicative function. To establish a lower bound for $S_{1/2}(\mathcal {M},a_{\mathbb {K}})$ , we need to use an auxiliary function $a_{\mathbb {K}}^\prime (n)$ . For the sake of completeness, we include the proof of Proposition 3.1 in the Appendix.

Remark. For $1/2 < \sigma <1$ , one can define analogously

(3-3) $$ \begin{align} S_{\sigma}(\mathcal{M},a_{\mathbb{K}}) {:=}q \sum_{m,n\in \mathcal{M}} a_{\mathbb{K}}\bigg(\frac{m}{(m,n)}\bigg)a_{\mathbb{K}}\bigg(\frac{n}{(m,n)}\bigg)\bigg(\frac{(m,n)}{[m,n]}\bigg)^{\sigma}. \end{align} $$

If a lower bound similar to that in Proposition 3.1 can be obtained for $S_{\sigma }(\mathcal {M},a_{\mathbb {K}})$ , the result of Theorem 1.2 can be optimized through a process nearly analogous to that of Theorem 1.1.

3.2 Constructing the resonator

Following the ideas from [Reference Bondarenko and Seip8, Reference de la Bretèche and Tenenbaum12, Reference Yang29], we construct the resonator $R(t)$ needed to apply the resonance method. For each integer $j \ge 0$ , define

$$ \begin{align*} \mathcal{N}_j{:=}q\bigg[\bigg(1 + \frac{\log T}{T}\bigg)^{j}, \bigg(1 + \frac{\log T}{T}\bigg)^{j + 1}\bigg)\cap\mathcal{M}. \end{align*} $$

Here, T is a parameter and $\mathcal {M}$ is defined as (3-2). Using the notation in [Reference de la Bretèche and Tenenbaum12], we let $h_j {:=}q \min \mathcal {N}_j$ if $\mathcal {N}_j \neq \emptyset $ . Define $\mathcal {H}$ as the set of all $h_j$ and consider the function r on $\mathcal {H}$ defined by

$$ \begin{align*} r(h_{j})=\sqrt{\sum_{m\in\mathcal{N}_j}1}. \end{align*} $$

We then define the resonator as follows:

$$ \begin{align*} R(t)=\sum_{h\in\mathcal{H}}r(h)h^{-it}. \end{align*} $$

By the Cauchy–Schwarz inequality,

$$ \begin{align*} |R(t)|^2 \le R(0)^{2}\le N\sum_{h\in\mathcal{H}}r(h)^{2}\le N|\mathcal{M}|\le N^{2}. \end{align*} $$

Set $\Phi (y){:=}q e^{-y^{2}/2}$ as in [Reference Bondarenko and Seip8]. Plainly, the Fourier transform $\widehat {\Phi }$ satisfies $\widehat {\Phi }(\xi )=\sqrt {2\pi }\Phi (\xi )$ . According to [Reference Bondarenko and Seip9, Lemma 5],

(3-4) $$ \begin{align} \int_{\mathbb{R}} |R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\, dt\ll\frac{|\mathcal{M}|T}{\log T}. \end{align} $$

3.3 The proof of Theorem 1.1

We now complete the proof of Theorem 1.1. Let $\eta = 2\phi (d)$ as in [Reference Bondarenko, Darbar, Hagen, Heap and Seip5] and $N = \lfloor T^\lambda \rfloor $ . For fixed $\varepsilon>0$ , we choose $\lambda $ such that $\lambda +5\varepsilon <1$ . We start with the following integral:

$$ \begin{align*} J(T){:=}q\int_{|t|\ge1}\int_{\mathbb{R}}\mathcal{G}\bigg(\frac{1}{2} + it + iu\bigg)\mathcal{G}\bigg(\frac{1}{2} - it + i u\bigg)K_{\eta}(u)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,du\,dt. \end{align*} $$

We divide $J(T)$ into three parts as follows:

(3-5)

For brevity, the integrands on the right-hand side are omitted, without causing any ambiguity. We show that the integral $J_1(T)$ provides the main term for $J(T)$ .

Using Lemma 2.1, we establish the following bound:

(3-6) $$ \begin{align} \bigg|\mathcal{G}\bigg(\frac{1}{2}\pm it + iu\bigg)\bigg|\ll 1 + \bigg|\zeta_{\mathbb{K}}^{(\ell)}\bigg(\frac{1}{2} \pm it + iu\bigg)\bigg|\ll (1 + |t| + |u|)^{\frac{3}{10}\phi(d)}. \end{align} $$

Combining with $K_{\eta }(u) \ll u^{-4\phi (d)},$ it is clear that

$$ \begin{align*} J_2 (T) &\,\ll d^{\phi(d)} \int_{1\le |t| \le T \log T}\int_{|u| \ge T^{1/2}} (1 + |t| + |u|)^{3\phi(d)/5} K_{\eta}(u)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,du\,dt \\ &\,\ll d^{\phi(d)} ( T \log T)^{3\phi(d)/5}\int_{|u| \ge T^{1/2}} |u| ^{3\phi(d)/5-4\phi(d)}\,du \int_{\mathbb{R}}|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt. \end{align*} $$

Then, by (3-4) and $d \ll (\log _2 T)^A,$

(3-7) $$ \begin{align} J_2(T) \ll d^{\phi(d)} |\mathcal{M}|T^{-7\phi(d)/20+\varepsilon}=o(|\mathcal{M}|T). \end{align} $$

Moreover,

$$ \begin{align*} J_3 (T) \ll d^{\phi(d)} \int_{|t| \ge T \log T} |t|^{3\phi(d)/5}|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt \int_{\mathbb{R}} |u|^{3\phi(d)/5} K_{\eta}(u)\,du. \end{align*} $$

The rapid decay of $\Phi $ and (3-6) implies that

(3-8) $$ \begin{align} J_3 \ll d^{\phi(d)}|\mathcal{M}| T^{3\phi(d)/5+1+\varepsilon} \Phi \bigg(\frac{(\log T)^2}{\sqrt{2}}\bigg)=o(|\mathcal{M}|T). \end{align} $$

Substituting (3-7), (3-8) into (3-5),

$$ \begin{align*} J(T) = J_1 (T) + o(|\mathcal{M}|T). \end{align*} $$

Define

$$ \begin{align*} \mathcal{Z} {:=}q \mathcal{Z}_{\mathbb{K}}(T)= \max_{-T \log T - T^{1/2}\le t\le T \log T+T^{1/2}}\bigg|\mathcal{G}\bigg(\frac{1}{2}+it\bigg)\bigg|. \end{align*} $$

Combining this with (3-4), we immediately derive the following lower bound:

$$ \begin{align*} J_1(T) \ll \widehat{K_{\eta}}(0) \mathcal{Z}^2 \frac{|\mathcal{M}|T}{\log T}. \end{align*} $$

Thus,

(3-9) $$ \begin{align} J(T) \ll \widehat{K_{\eta}}(0) \mathcal{Z}^2 \frac{|\mathcal{M}|T}{\log T} + o (|\mathcal{M}|T). \end{align} $$

Next, we seek an effective lower bound for $J(T)$ . Our approach establishes a connection between $J(T)$ and GCD sums via the double-version convolution formula. We introduce the following three integrals:

$$ \begin{align*} &J_0(T){:=}q \int_{|t|\ge 1}E(t)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt, \\ &\mathcal{E}^+{:=}q -\int_{|t|\ge 1}\tau^{+}\cdot|R(t)|^{2}\Phi\bigg( \frac{t\log T}{T}\bigg)\,dt, \\ &\mathcal{E}^-{:=}q -\int_{|t|\ge 1}\tau^{-}\cdot|R(t)|^{2}\Phi\bigg( \frac{t\log T}{T}\bigg)\,dt, \end{align*} $$

where the series $E(t)$ is defined as

$$ \begin{align*} E(t){:=}q \sum_{m,n\ge1}\frac{\widehat{K_{\eta}}(\log nm)}{\sqrt{nm}(n/m)^{it}}a(n)a(m). \end{align*} $$

Lemma 2.2 implies that $J(T)=J_0(T)+\mathcal {E}^+ +\mathcal {E}^-.$ As in [Reference Yang30, pp. 14–15],

$$ \begin{align*} |\mathcal{E}^+|\ll|\mathcal{M}| \frac{T^{\lambda}}{\log T} \quad \text{and} \quad |\mathcal{E}^-|\ll|\mathcal{M}| \frac{T^{\lambda}}{\log T}. \end{align*} $$

Subsequently, we use the Fourier transform on the real axis and, therefore, focus on the region $|t| \le 1$ now. Lemma 2.4 shows that $\widehat {K_{\eta }}(\log nm)=0$ when $mn\ge T^{2\varepsilon }$ . Thus,

$$ \begin{align*} \int_{\mathbb{R}} E(t)|R(t)|^2\Phi\bigg(\frac{t\log T}{T} \bigg) \, dt \ll \widehat{K_\eta}(0)R(0)^2\sum_{m,n\ge1}\frac{a(n)a(m)}{\sqrt{mn}}. \end{align*} $$

Since $d \ll (\log _2 T)^A,$ we have $a(n)n^{-1/2}\ll (\log _2 T)^A.$ Thus, the integral on $|t| \le 1$ can be bounded by

$$ \begin{align*} \widehat{K_{\eta}}(0)R(0)^2(\log_2 T)^A \sum_{mn\le T^{2\varepsilon}}1 \ll \widehat{K_{\eta}}(0)|\mathcal{M}|T^{\lambda + 5\varepsilon}. \end{align*} $$

We extend the integral to $\mathbb {R}$ . Specifically, setting

$$ \begin{align*} \widetilde{J}(T){:=}q\int_{\mathbb{R}} E(t)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt, \end{align*} $$

we have

$$ \begin{align*} J_0(T) = \widetilde{J}(T) + O(|\mathcal{M}|T^{\lambda + 5\varepsilon}). \end{align*} $$

Hence, combining with (3-9),

(3-10) $$ \begin{align} \widetilde{J}(T) \ll \widehat{K_{\eta}}(0) \mathcal{Z}^2 \frac{|\mathcal{M}|T}{\log T} + o(|\mathcal{M}| T) + O(|\mathcal{M}|T^{\lambda + 5\varepsilon}). \end{align} $$

We now derive the lower bound for $\widetilde {J}(T)$ . Expanding $|R(t)|^2$ and $E(t)$ in $\widetilde {J}(T)$ yields that

$$ \begin{align*} \widetilde{J}(T)=\sqrt{2\pi}\frac{T}{\log T}\sum_{g,k\geq1}\frac{\widehat{K_{\eta}}(\log gk)a(g)a(k)}{\sqrt{gk}}\sum_{h,h^{\prime}\in\mathcal{H}}r(h)r(h^{\prime})\Phi\bigg(\frac{T}{\log T}\log\frac{hg}{h^{\prime}k}\bigg). \end{align*} $$

For the terms involving $\widehat {K_{\eta }}(u)$ , we follow the approach in [Reference Bondarenko, Darbar, Hagen, Heap and Seip5, p. 10], using (2-3) and (2-4). For the inner sum, it follows from the proof of [Reference de la Bretèche and Tenenbaum12, Theorem 1.4] that

$$ \begin{align*} \widetilde{J}(T) \gg \widehat{K_{\eta}}(0)\frac{T}{\log T}\sum_{\substack{m,n\in\mathcal{M} \\ mg = nk}}\sum_{1\le gk\le T^{\varepsilon/3\eta}}\frac{a_{\mathbb{K}}(g)a_{\mathbb{K}}(k)}{\sqrt{gk}}. \end{align*} $$

Setting

$$ \begin{align*} g = \frac{n}{(m,n)}, \quad k = \frac{m}{(m,n)}, \quad gk = \frac{[m,n]}{(m,n)} \le T^{\varepsilon/3\eta}, \end{align*} $$

we have

$$ \begin{align*} \widetilde{J}(T) \gg \widehat{K_{\eta}}(0)\frac{T}{\log T}\sum_{\substack{m,n\in\mathcal{M}\\ [m,n]/(m,n)\le T^{\varepsilon/3\eta}}}a_{\mathbb{K}}\bigg(\frac{m}{(m,n)}\bigg)a_{\mathbb{K}}\bigg(\frac{n}{(m,n)}\bigg)\sqrt{\frac{(m,n)}{[m,n]}}. \end{align*} $$

Using Rankin’s trick, we can establish a connection between $\widetilde {J}$ and weighted GCD sums as follows:

$$ \begin{align*} \widetilde{J}(T)\gg \widehat{K_{\eta}}(0)\frac{T}{\log T}(S_{1/2}(\mathcal{M}, a_{\mathbb{K}}) - T^{-\varepsilon/18\eta}S_{1/3}(\mathcal{M},a_{\mathbb{K}})), \end{align*} $$

where $S_{1/2}(\mathcal {M},a_{\mathbb {K}})$ and $S_{1/3}(\mathcal {M},a_{\mathbb {K}})$ are given by (3-1) and (3-3), respectively.

Combining with (3-10),

$$ \begin{align*} \mathcal{Z}^2 \gg \frac{S_{1/2}(\mathcal{M}, a_{\mathbb{K}})}{|\mathcal{M}|} - \frac{S_{1/3}(\mathcal{M}, a_{\mathbb{K}})}{|\mathcal{M}|T^{\varepsilon/18\eta}} + o(\log T). \end{align*} $$

Following the idea from [Reference de la Bretèche and Tenenbaum12], if we set $P^+ (m)$ as the largest prime factor of m and $y_{\mathcal {M}} {:=}q \max _{m \in \mathcal {M}} P^+(m),$ then,

$$ \begin{align*}\frac{S_{1/3}(\mathcal{M}, a_{\mathbb{K}})}{|\mathcal{M}|} \ll \exp \big(\phi(d) y_{\mathcal{M}}^{2/3}\big).\end{align*} $$

Based on the construction of $P_k$ in Section 2, we have $y_{\mathcal {M}} \ll (\log T)^{6/5}$ , so that for sufficiently large T,

$$ \begin{align*} \frac{S_{1/3}(\mathcal{M}, a_{\mathbb{K}})}{T^{\varepsilon/18\eta}|\mathcal{M}|} \ll \frac{\exp\big(\phi(d) y_{\mathcal{M}}^{2/3}\big)}{T^{\varepsilon/18\eta}} \to 0. \end{align*} $$

Thus, by Proposition 3.1, for all sufficiently large T,

$$ \begin{align*} \mathcal{Z}^2 \gg \exp \bigg(\bigg(2\phi(d)\sqrt{\lambda}+o(1)\bigg)\sqrt{\frac{\log T\log_3 T}{\log_2 T}}\bigg). \end{align*} $$

Trivially,

(3-11) $$ \begin{align} \max_{-T \log T - T^{1/2}\le t\le T \log T +T^{1/2}}&\,\bigg|\zeta_{\mathbb{K}}^{(\ell)}\bigg(\frac{1}{2}+it\bigg)\bigg| \gg \mathcal{Z} + O(1) \nonumber \\ &\, \gg \exp\bigg(\bigg(\phi(d)\sqrt{\lambda}+o(1)\bigg)\sqrt{\frac{\log T\log_3 T}{\log_2 T}}\bigg). \end{align} $$

We let $\lambda $ be sufficiently close to 1 such that $\lambda + 5\varepsilon <1$ for small $\varepsilon $ . Finally, we set

$$ \begin{align*}T^{\prime}=(1+o(1))\frac{T}{\log T}\end{align*} $$

and readjust the parameter $T=T^{\prime }$ following the idea from [Reference Bondarenko, Darbar, Hagen, Heap and Seip5]. Variations in the logarithmic factor affect only the lower order terms in the exponent on the right-hand side of (3-11). Hence, this result also holds for $t \in [0,T]$ . We complete the proof of Theorem 1.1.

4.1 Constructing the resonator

We construct the resonator $R(t)$ following a similar strategy as in [Reference Chirre11]. Let $\gamma \in (0,1)$ be a parameter to be chosen later. Define $\mathcal {P}_d$ as the set of prime numbers p such that

$$ \begin{align*} e \phi(d) \log N \log_{2}N < p \le \phi(d)\log N\exp((\log_2 N)^{\gamma})\log_{2}N \quad \text{and} \quad p\equiv 1 \quad\pmod{\ d}. \end{align*} $$

For any $1 \le c_d \le \sqrt {\phi (d)/(e-1)}$ , define the multiplicative function $f(n)$ supported on the set of square-free numbers by setting

$$ \begin{align*} f(p) = c_d \bigg( \frac{(\log N)^{1 - \sigma} (\log_2 N)^{\sigma}}{(\log_3 N)^{1 - \sigma}} \bigg) \frac{1}{p^{\sigma} (\log p - \log_2 N - \log_3 N - \log \phi(d))} \end{align*} $$

as $p \in \mathcal {P}_d$ , and $f(p) = 0$ otherwise.

For each integer $k \in \{1, \ldots , \lfloor (\log _2 N)^{\gamma }\rfloor \}$ , define $\mathcal {P}_{k,d}$ as the set of p satisfying

$$ \begin{align*} e^k \phi(d) \log N \log_{2}N < p \le e^{k+1}\phi(d)\log N\log_{2}N \quad \text{and} \quad p\equiv 1 \quad\pmod{\ d}, \end{align*} $$

which is the subset of $\mathcal {P}_d$ . Fixing $1 < \alpha < 1/\gamma $ , define

$$ \begin{align*} \mathcal{M}_{k,d} {:=}q \bigg\{ n \in \mathrm{supp}(f) : n \text{ has at least } \Delta_{k} {:=}q \frac{\alpha (\log N)^{2 - 2\sigma}}{k^2 (\log_3 N)^{2 - 2\sigma}} \text{ prime divisors in } \mathcal{P}_{k,d} \bigg\}. \end{align*} $$

Let $\mathcal {M}^{\prime }_{k,d}$ consist of those integers $\mathcal {M}_{k,d}$ whose prime factors all lie in $\mathcal {P}_{k,d}$ . Subsequently, set

$$ \begin{align*} \mathcal{M}_d {:=}q \mathrm{supp}(f) \setminus \bigcup_{k = 1}^{\lfloor(\log_2 N)^{\gamma}\rfloor} \mathcal{M}_{k,d}. \end{align*} $$

From the above definitions, we can deduce several useful properties. First, $\mathcal {M}_d$ is divisor closed. Specifically, if some $m^{\prime } \mid m \in \mathcal {M}_d,$ then $m^{\prime } \in \mathcal {M}_d$ . Adapting the argument of [Reference Chirre11, Lemma 8], we obtain that $|\mathcal {M}_d| \le N$ . Furthermore, the Siegel–Walfisz theorem gives

(4-1) $$ \begin{align} |\mathcal{P}_{k,d}| \le (1 + o(1)) e^{k+1} \log N. \end{align} $$

Now, we construct the resonator $R(t)$ . Define $\mathcal {J}$ as the set of integers j such that

$$ \begin{align*} \bigg[\bigg(1 + \frac{\log T}{T}\bigg)^j, \bigg(1 + \frac{\log T}{T}\bigg)^{j + 1}\bigg] \bigcap \mathcal{M}_d \neq \emptyset, \end{align*} $$

and let $m_j$ be the minimum of $[(1 + \log T/T)^j, (1 + \log T/T)^{j + 1}] \cap \mathcal {M}_d $ for all $j \in \mathcal {J}.$ Setting $\mathcal {M}^{\prime }_d = \{m_j : j \in \mathcal {J} \}$ , define

$$ \begin{align*} r(m_j) {:=}q \bigg( \sum_{n \in \mathcal{M}_d, (1 - \log T/T)^{j - 1} \le n \le (1 + \log T/T)^{j + 2}} f(n)^2 \bigg)^{1/2}. \end{align*} $$

Finally, define

$$ \begin{align*} R(t) {:=}q \sum_{m \in \mathcal{M}^{\prime}_d} r(m) m^{-it}. \end{align*} $$

By the Cauchy–Schwarz inequality,

$$ \begin{align*} |R(t)|^2 \le R(0)^2 \ll N \sum_{n \in \mathbb{N}} f(n)^2. \end{align*} $$

Moreover, taking $\Phi $ to be the same as in Section 3 and using [Reference Bondarenko and Seip9, Lemma 5],

$$ \begin{align*} \int_{\mathbb{R}} |R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt \ll \frac{T}{\log T}\sum_{n \in \mathbb{N}} f(n)^2. \end{align*} $$

4.2 Supporting propositions

Define

$$ \begin{align*} \mathcal{A}_d &{:=}q \mathcal{A}_{d,N}= \frac{1}{\sum_{n\in \mathbb{N}} f(n)^2} \sum_{n\in\mathbb{N}} \frac{f(n)}{n^{\sigma}} \sum_{q\mid n} a_{\mathbb{K}}\bigg(\frac{n}{q}\bigg)f(q)q^{\sigma} \\ & = \prod_{p\in \mathcal{P}_d} \frac{1 + f(p)^2 + \phi(d)f(p)p^{-\sigma}}{1 + f(p)^2}. \end{align*} $$

Here, $a_{\mathbb {K}}(n)$ are the coefficients of the Dirichlet series of $\zeta _{\mathbb {K}}(s)$ and f is the multiplicative function introduced in Section 4.1. In this subsection, we present some auxiliary propositions that provide a sharp lower bound for $\mathcal {A}_d$ and show that the error terms caused by truncation or the exclusion of $\mathcal {M}_d$ are negligible. These results play a central role in establishing the lower bound in Theorem 1.2.

Proposition 4.1. Let $d \ll (\log _2 N)^A$ and $c_d \le \sqrt {\phi (d)/(e-1)}$ . Then, for $0 < \delta <1$ as $N \to +\infty $ ,

$$ \begin{align*}\mathcal{A}_d \geq \exp\bigg((\delta\gamma c_d+ o(1))\frac{(\log N)^{1-\sigma}(\log_3 N)^{\sigma}}{(\log_2 N)^{\sigma}}\bigg).\end{align*} $$

Proof. For $c_d \le \sqrt {\phi (d)/(e-1)}$ , we have $f(p)=o(1)$ for all $p \in \mathcal {P}_d$ . It follows that

$$ \begin{align*} \mathcal{A}_d = \exp \bigg((1+o(1)) \sum_{p \in \mathcal{P}_d} \frac{f(p)}{p^{\sigma}}\bigg). \end{align*} $$

By the definition of $f(p)$ ,

$$ \begin{align*} \sum_{p \in \mathcal{P}_d} \frac{f(p)}{p^{\sigma}} & =c_d \frac{(\log N)^{1 - \sigma} (\log_2 N)^{\sigma}}{(\log_3 N)^{1 - \sigma}} \sum_{p \in \mathcal{P}_d} \frac{1}{p^{2\sigma}(\log p-\log _2 N-\log _3 N - \log \phi(d))} \\ & \ge c_d \frac{(\log N)^{1 - \sigma} (\log_2 N)^{\sigma}}{(\log_3 N)^{1 - \sigma}} \sum_{k=1}^{\lfloor(\log_2 N)^{\gamma}\rfloor} \sum_{p \in \mathcal{P}_{k,d}} \frac{1}{(k+1)p^{2\sigma}}. \end{align*} $$

Notice that from [Reference Chirre11, Lemma 9], we get the following lower bound:

$$ \begin{align*} \sum_{p \in \mathcal{P}_{k,d}} \frac{1}{p^{2\sigma}}> (\delta + o(1))(\log_2 N)^{ -2\sigma}, \end{align*} $$

where $0< \delta <1$ . Thus,

$$ \begin{align*} \sum_{p \in \mathcal{P}_d} \frac{f(p)}{p^{\sigma}} \ge (\delta\gamma c_d + o(1))\frac{(\log N)^{1-\sigma} (\log_3 N)^{\sigma}}{(\log_2 N)^{\sigma}}. \end{align*} $$

This completes the proof.

Proposition 4.2. Let $d \ll (\log _2 N)^A$ and $c_d \le \sqrt {\phi (d)/(e-1)}$ . Then, as $N \to +\infty $ ,

$$ \begin{align*} \frac{1}{\sum_{n \in \mathbb{N}} f(n)^2} \sum_{n \in \mathbb{N}, n \notin \mathcal{M}_d} \frac{f(n)}{n^{\sigma}} \sum_{q \mid n} a_{\mathbb{K}} \bigg( \frac{n}{q}\bigg)f(q)q^{\sigma}=o(\mathcal{A}_d). \end{align*} $$

Proof. By the definition of $\mathcal {M}_d,$

$$ \begin{align*} \mathcal{F} &{:=}q \frac{1}{\mathcal{A}_d \sum_{n\in\mathbb{N}} f(n)^2} \sum_{n\in\mathbb{N}, n\notin\mathcal{M}_d} \frac{f(n)}{n^{\sigma}} \sum_{q\mid n}a_{\mathbb{K}} \bigg( \frac{n}{q}\bigg)f(q)q^{\sigma} \\ &\le \frac{1}{\mathcal{A}_d \sum_{n\in\mathbb{N}} f(n)^2} \sum_{k = 1}^{\lfloor(\log_2 N)^{\gamma}\rfloor} \sum_{n\in\mathcal{M}_{k,d}} \frac{f(n)}{n^{\sigma}} \sum_{q\mid n} a_{\mathbb{K}} \bigg( \frac{n}{q}\bigg)f(q)q^{\sigma}. \end{align*} $$

From the construction of $f(n)$ and $\mathcal {M}_{k,d}^{\prime }$ , it follows that $\mathcal {F}$ can be bounded by

It is clear from (4-1) that

$$ \begin{align*} \prod_{p\in \mathcal{P}_{k,d}} \bigg(1 + \frac{\phi(d)}{f(p)p^{\sigma}}\bigg)& \le \prod_{p\in \mathcal{P}_{k,d}} \bigg(1 + \frac{\phi(d)}{c_d}(k+1)\frac{(\log_3 N)^{1-\sigma}}{(\log N)^{1-\sigma}(\log_2 N)^{\sigma}}\bigg) \\ &\le \exp\bigg(\frac{\phi(d)}{c_d}(k + 1)e^{k + 1}\frac{(\log N)^{\sigma}(\log_3 N)^{1-\sigma}}{(\log_2 N)^{\sigma}}\bigg). \end{align*} $$

Therefore, combining this with the fact that $d \ll ( \log _2 N)^A$ and $k \le (\log _2 N)^{\gamma }$ ,

$$ \begin{align*} \prod_{p\in \mathcal{P}_{k,d}} \bigg(1 + \frac{\phi(d)}{f(p)p^{\sigma}}\bigg) =\exp\bigg(o\bigg(\frac{(\log N)^{2-2\sigma}}{k^2 (\log_3 N)^{2-2\sigma}}\bigg)\bigg). \end{align*} $$

However, recalling the definition of $\mathcal {M}^{\prime }_{k,d},$ we use the classical method in [Reference Bondarenko and Seip8, p. 8]. Since $f(n)$ is multiplicative,

$$ \begin{align*} \sum_{n\in\mathcal{M}^{\prime}_{k,d}} f(n)^2 \le b^{-\Delta_k} \prod_{p\in \mathcal{P}_{k,d}} (1 + b f(p)^2) \end{align*} $$

whenever $b>1$ . Thus,

$$ \begin{align*} \frac{1}{\prod_{p\in \mathcal{P}_{k,d}} (1 + f(p)^2)} \sum_{n\in\mathcal{M}^{\prime}_{k,d}} f(n)^2 \le b^{-\Delta_k} \exp\bigg( (b - 1)\sum_{p\in \mathcal{M}^{\prime}_{k,d}}f(p)^2\bigg). \end{align*} $$

For the sum in the exponent, the Siegel–Walfisz theorem yields

$$ \begin{align*} \sum_{p\in \mathcal{M}^{\prime}_{k,d}} f(p)^2 &= c_d^2\frac{(\log N)^{2-2\sigma}(\log_2 N)^{2\sigma}}{(\log_3 N)^{2-2\sigma}} \sum_{p\in \mathcal{P}_{k,d}} \frac{1}{p^{2\sigma}(\log p - \log_2 N - \log_3 N - \log \phi(d))^2} \\ &\le \frac{c_d^2}{\phi(d)} (1 + o(1))\frac{(\log N)^{2-2\sigma}(\log_2 N)^{2\sigma}}{(\log_3 N)^{2-2\sigma}} \int_{e^k \phi(d) \log N \log_2 N}^{e^{k+1} \phi(d) \log N \log_2 N} \frac{dx}{k^2 x^{2\sigma}\log x}\\ &\le \frac{c_d^2 (e-1)}{\phi(d)}(1 + o(1))\frac{(\log N)^{2-2\sigma}}{k^2 (\log_3 N)^{2-2\sigma}}. \end{align*} $$

Combining all the above estimates, $\mathcal {F}^\star $ is bounded by

$$ \begin{align*} \exp\bigg(\bigg(\frac{c_d^2 (e-1)}{\phi(d)}(b - 1) - \alpha\log b + o(1)\bigg)\frac{(\log N)^{2-2\sigma}}{k^2 (\log_3 N)^{2-2\sigma}}\bigg). \end{align*} $$

Since $c_d \le \sqrt {\phi (d)/(e-1)}$ and $\alpha>1$ , it suffices to choose b sufficiently close to $1$ so that the exponent becomes negative. Hence, the proof is complete.

Proposition 4.3. Let $d \ll (\log _2 N)^A$ and $c_d \le \sqrt {\phi (d)/(e-1)}$ . Then, as $N \to +\infty $ ,

$$ \begin{align*} \frac{1}{\sum_{n \in \mathbb{N}} f(n)^2} \sum_{n \in \mathcal{M}_d} \frac{f(n)}{n^{\sigma}} \sum_{q \mid n, q \leq n/N^{\varepsilon/3\eta}} a_{\mathbb{K}} \bigg( \frac{n}{q}\bigg)f(q)q^{\sigma}=o(\mathcal{A}_d). \end{align*} $$

Proof. We have

$$ \begin{align*} \sum_{n \in \mathcal{M}_d} \frac{f(n)}{n^{\sigma}} \sum_{q \mid n, q \leq n/N^{\varepsilon/3\eta}} a_{\mathbb{K}} \bigg( \frac{n}{q}\bigg)f(q)q^{\sigma} = \sum_{n \in \mathcal{M}_d} f(n)^2 \sum_{q \mid n, q \geq N^{\varepsilon/3\eta}} \frac{a_{\mathbb{K}}(q)}{f(q)q^\sigma}. \end{align*} $$

Furthermore, applying Rankin’s trick to the inner sum on the right-hand side yields

$$ \begin{align*} \sum_{q \mid n, q \geq N^{\varepsilon/3\eta}} \frac{a_{\mathbb{K}}(q)}{f(q)q^\sigma} \le N^{-\varepsilon/12\eta}\sum_{q \mid n} \frac{a_{\mathbb{K}}(q)}{f(q)q^{\sigma-1/4}} = N^{-\varepsilon/12\eta} \prod_{p \mid n}\bigg(1+ \frac{\phi(d)}{f(p)p^{\sigma-1/4}} \bigg). \end{align*} $$

Since $\eta = 2\phi (d)$ and because each n in $\mathcal {M}_d$ has at most $ O((\log N / \log _3 N)^{2-2\sigma })$ prime factors, the proof follows by a process similar to that in [Reference Bondarenko, Darbar, Hagen, Heap and Seip5, Lemma 6].

4.3 The proof of Theorem 1.2

In this subsection, the ideas and techniques in our preceding part are essentially the same as those used in Section 3.3. Let $N=\lfloor T^\lambda \rfloor $ , where $\lambda + 2 \varepsilon < 1$ holds for small $\varepsilon $ . Define

$$ \begin{align*} W_1(T) {:=}q \int_{1\le |t| \le T \log T}\int_{ |u| \le T^{1/2}}\mathcal{G}(\sigma + it + iu)K_{\eta}(u)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,du\,dt \end{align*} $$

and

$$ \begin{align*} \mathcal{Z}(\sigma) {:=}q \mathcal{Z}_{\mathbb{K}}(\sigma,T) = \max_{-T \log T - T^{1/2}\le t\le T \log T+T^{1/2}}|\mathcal{G}(\sigma+it)|. \end{align*} $$

Trivially,

$$ \begin{align*} W_1(T) \le \widehat{K_{\eta}}(0)\mathcal{Z}(\sigma) \int_{\mathbb{R}} |R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt \ll \widehat{K_{\eta}}(0)\mathcal{Z}(\sigma)\frac{T}{\log T}\sum_{n \in \mathbb{N}} f(n)^2. \end{align*} $$

Then, we proceed by considering the tails of the integrals:

$$ \begin{align*} W_2(T) &\, {:=}q\int_{1\le |t| \le T \log T}\int_{|u| \ge T^{1/2}}\mathcal{G}(\sigma+it+iu)K_{\eta}(u)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,du\,dt, \\ W_3(T) &\, {:=}q\int_{|t| \ge T \log T}\int_{\mathbb{R}}\mathcal{G}(\sigma + it + iu)K_{\eta}(u)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,du\,dt. \end{align*} $$

The following upper bound for $W_2(T)$ can be established by using Lemma 2.1 and the fact that $K_{\eta }(u) \ll u^{-4\phi (d)}$ ,

(4-2) $$ \begin{align} W_2(T) & \ll d^{\phi(d)} \int_{1\le |t| \le T \log T}\int_{|u| \ge T^{1/2}} (1 + |t| + |u|)^{3\phi(d)/10} K_{\eta}(u)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,du\,dt \nonumber \\ & \ll d^{\phi(d)}(T\log T)^{3\phi(d)/10} T^{(3\phi(d)/10 - 4\phi(d)+1)/2} \int_{\mathbb{R}}|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt \nonumber \\ & = o\bigg(\frac{T}{\log T} \sum_{n \in \mathbb{N}} f(n)^2\bigg). \end{align} $$

Furthermore, for $W_3(T)$ , applying Lemma 2.1,

$$ \begin{align*} W_3(T) \ll d^{\phi(d)} \int_{|t| \ge T\log T}|t|^{3\phi(d)/10}|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt \int_{\mathbb{R}} |u|^{3\phi(d)/10} K_{\eta}(u)\,du. \end{align*} $$

Since $\Phi $ decays rapidly,

(4-3) $$ \begin{align} W_3(T) & \ll d^{\phi(d)} (T\log T)^{3\phi(d)/10} \Phi \bigg(\frac{(\log T)^2}{\sqrt{2}}\bigg) \int_{\mathbb{R}} |R(t)|^{2} \Phi \bigg(\frac{t\log T}{\sqrt{2}T}\bigg)\,dt \nonumber \\ & = o \bigg(\frac{T}{\log T} \sum_{n \in \mathbb{N}} f(n)^2\bigg). \end{align} $$

Equations (4-2) and (4-3) yield the following result:

(4-4) $$ \begin{align} W(T) = W_1(T) + o\bigg(\frac{T}{\log T} \sum_{n \in \mathbb{N}} f(n)^2\bigg) \ll (\widehat{K_{\eta}}(0)\mathcal{Z}(\sigma)+o(1))\frac{T}{\log T}\sum_{n \in \mathbb{N}} f(n)^2, \end{align} $$

where

$$ \begin{align*} W(T) {:=}q \int_{|t|\ge1}\int_{\mathbb{R}}\mathcal{G}(\sigma + it + iu)K_{\eta}(u)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,du\,dt. \end{align*} $$

Then, define

$$ \begin{align*} W_0(T) &{:=}q \int_{|t| \ge 1}\bigg(\sum_{n=1}^{\infty}\frac{\widehat{K_{\eta}}(\log n)a(n)}{n^{\sigma +it}}\bigg)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt, \\ \mathcal{E} &{:=}q \int_{|t| \ge1} 2\pi \tau \cdot |R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt. \end{align*} $$

Cauchy’s integral formula implies $\mathcal {E} = o(T^{\lambda } \sum _{n \in \mathbb {N}}f(n)^2)$ , and thus,

(4-5) $$ \begin{align} W(T) = W_0(T) + o\bigg(T^{\lambda} \sum_{n \in \mathbb{N}}f(n)^2\bigg). \end{align} $$

To extend the t integral to $\mathbb {R}$ , we consider the integral over the region $|t| \le 1$ . Plainly,

$$ \begin{align*} \mathcal{E}^{\prime}{:=}q\int_{|t| \le 1}\bigg(\sum_{n=1}^{\infty}\frac{\widehat{K_{\eta}}(\log n)a(n)}{n^{\sigma +it}}\bigg)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt \ll R(0)^{2} \sum_{n=1}^{\infty}\frac{\widehat{K_{\eta}}(\log n)a(n)}{n^{\sigma}}. \end{align*} $$

Combining this with Lemma 2.4, we deduce that

(4-6) $$ \begin{align} \mathcal{E}^{\prime}\ll R(0)^{2} \widehat{K_{\eta}}(0) (\log_2 T)^A \sum_{n\leqslant T^{2\varepsilon}}1 \ll \widehat{K_{\eta}}(0) T^{\lambda + 2 \varepsilon} \sum_{n \in \mathbb{N}}f(n)^2. \end{align} $$

Recalling that $\lambda + 2\varepsilon <1$ and combining (4-4), (4-5), and (4-6), it follows that

$$ \begin{align*} \mathcal{Z}(\sigma) \gg \int_{\mathbb{R}}\bigg(\sum_{n=1}^{\infty}\frac{\widehat{K_{\eta}}(\log n)a_{\mathbb{K},\ell}(n)}{n^{\sigma +it}}\bigg)|R(t)|^{2}\Phi\bigg(\frac{t\log T}{T}\bigg)\,dt + o(1). \end{align*} $$

Denoting $\widetilde {W}(T)$ as the integral on the right-hand side of the preceding inequality,

$$ \begin{align*} \widetilde{W}(T)= \sqrt{2 \pi} \frac{T}{\log T} \sum_{m,n \in \mathcal{M}^{\prime}_d} \sum_{k=1}^{\infty}\frac{\widehat{K_{\eta}}(\log n)a_{\mathbb{K}}(k)r(m)r(n)}{k^{\sigma}} \Phi\bigg(\frac{T}{\log T} \log \frac{km}{n}\bigg). \end{align*} $$

Here, we note the fact that $a(n) \ge a_{\mathbb {K}}(n)$ . Following the idea of [Reference Bondarenko, Darbar, Hagen, Heap and Seip5], we restrict the inner sum to $k \le T^{\varepsilon /3\eta }$ , yielding

$$ \begin{align*} \widetilde{W}(T) \gg \widehat{K_{\eta}}(0) \frac{T}{\log T} \sum_{m,n\in\mathcal{M}^{\prime}_d} \sum_{k\le T^{\varepsilon/3n}} \frac{a(k)r(m)r(n)}{k^{\sigma}} \Phi\bigg(\frac{T}{\log T} \log \frac{km}{n}\bigg). \end{align*} $$

Employing techniques similar to those in [Reference Bondarenko and Seip9, (21)],

$$ \begin{align*} \widetilde{W}(T) & \gg \widehat{K_{\eta}}(0) \frac{T}{\log T}\sum_{m,n\in\mathcal{M}_d} \sum_{\substack{km = n \\k\le T^{\varepsilon/3\eta}}} \frac{a_{\mathbb{K}}(k)f(m)f(n)}{k^{\sigma}} \nonumber \\ & = \widehat{K_{\eta}}(0) \frac{T}{\log T} \sum_{n\in\mathcal{M}_d} \frac{f(n)}{n^{\sigma}} \sum_{q\mid n, q\ge n/T^{\varepsilon/3\eta}} a_{\mathbb{K}}\bigg(\frac{n}{q}\bigg)f(q)q^{\sigma}. \end{align*} $$

Combining Propositions 4.1, 4.2, and 4.3 yields

$$ \begin{align*}\mathcal{Z}(\sigma) \gg \exp\bigg((\delta\gamma c_d + o(1))\frac{(\log T )^{1-\sigma}(\log_3 T)^{\sigma}}{(\log_2 T)^{\sigma}}\bigg) + o(1)\end{align*} $$

for $\lambda + 2\varepsilon <1.$ Then,

$$ \begin{align*} \max_{-T \log T - T^{1/2}\le t\le T \log T +T^{1/2}}|\zeta_{\mathbb{K}}^{(\ell)}(\sigma+it)| & \gg \mathcal{Z}(\sigma) + O(1)\\ &\gg \exp\bigg((\delta\gamma c_d + o(1))\frac{(\log T)^{1-\sigma} (\log_3 T)^{\sigma}}{(\log_2 T)^{\sigma}}\bigg) + o(1). \end{align*} $$

Let $\lambda +2 \varepsilon <1$ with small $\varepsilon $ . By choosing $\delta $ , $\gamma $ , and $c_d$ to be sufficiently close respectively to $1$ , $1$ , and $\sqrt {\phi (d)/(e-1)}$ , and applying the same transform as in Section 3.3, we conclude the proof of Theorem 1.2.