2 Preliminaries

In this section, we carry out some preliminary work. We begin with the following asymptotic formula, which shows that derivatives of Dirichlet L-functions can be effectively approximated by a truncated Dirichlet series.

Lemma 2.1. Fix $\ell \in \mathbb {N}_+$ and $\sigma \in (1/2,1)$ . When q is sufficiently large, for any nonprincipal character $\chi \pmod {q}$ ,

$$ \begin{align*} (-1)^\ell L^{(\ell)}(\sigma,\chi) = \sum_{n\le q}\frac{(\log n )^\ell \chi(n)}{n^\sigma} + O_{\sigma,\ell} ( q^{({1}/{2})-\sigma}(\log q)^{\ell+1}). \end{align*} $$

Proof. Trivially,

(2.1) $$ \begin{align} (-1)^\ell L^{(\ell)}(\sigma,\chi) = \sum_{n \in \mathbb{N}} \frac{(\log n )^\ell \chi(n)}{n^\sigma} = \bigg(\sum_{n\le q} +\sum_{n> q}\bigg) \frac{(\log n )^\ell \chi(n)}{n^\sigma}. \end{align} $$

Using partial summation together with the Pólya–Vinogradov inequality,

$$ \begin{align*} \bigg|\sum_{n> q}\frac{(\log n)^\ell\chi(n)}{n^\sigma}\bigg| &\ll_{\sigma,\ell} \sqrt{q}\log q \bigg( \frac{(\log q)^\ell}{q^\sigma}+\int_q^\infty \frac{(\log t)^\ell}{t^{\sigma+1}}\, dt\bigg) \\ & \ll_{\sigma,\ell} q^{(1/2)-\sigma}(\log q)^{\ell+1}. \end{align*} $$

Substituting this into (2.1) completes the proof of Lemma 2.1.

Next, we construct the resonator $R(\chi )$ . Define $Y= \eta \log q \log \log q$ , where $0 < \eta < (2\log 4)^{-1} \sigma $ . As in [Reference Aistleitner, Mahatab, Munsch and Peyrot3], define $r(n)$ to be a completely multiplicative function whose values at primes p are given by

$$ \begin{align*} r(p)= \begin{cases} 1/2, & p\leq Y,\\ 0, & p>Y. \end{cases} \end{align*} $$

Furthermore, define the resonator

$$ \begin{align*} R(\chi) = \sum_{n \in \mathbb{N}}r(n)\, \chi(n). \end{align*} $$

Since $r(n)$ is multiplicative, $R(\chi )$ can be written in the Euler product form

$$ \begin{align*} R(\chi) = \prod_{p\le Y} (1-r(p)\,\chi(p) )^{-1} = \prod_{p\le Y} \bigg(1-\frac{\chi(p)}{2} \bigg)^{-1}. \end{align*} $$

3 Proof of Theorem 1.1

In this section, we prove Theorem 1.1 using the long resonator method. Let $\mathcal {G}_q$ denote the set of all Dirichlet characters $\chi $ modulo q. Since $\chi (q)=0$ for every Dirichlet character modulo q, the term $n=q$ in Lemma 2.1 vanishes. We therefore define

$$ \begin{align*} P(\chi) : = P_\ell(\chi) = \sum_{n< q}\frac{(\log n )^\ell \chi(n)}{n^\sigma}. \end{align*} $$

Furthermore, define the two sums

$$ \begin{align*} S_1 := S_1(R,q) = \sum_{\chi \in \mathcal{G}_q}|R(\chi)|^2, \quad S_2 := S_2(R,q) = \sum_{\chi \in \mathcal{G}_q}P(\chi) |R(\chi)|^2. \end{align*} $$

For $S_1$ , we expand $|R(\chi )|^2$ and apply the orthogonality of Dirichlet characters, which gives

(3.1) $$ \begin{align} S_1 = \sum_{\chi \in \mathcal{G}_q}\, \sum_{m,n \ge 1} r(m)r(n)\chi(m)\overline\chi(n) = \phi(q)\sum_{m \equiv n \pmod q} r(m)r(n), \end{align} $$

where $\phi (n)$ is Euler’s totient function. As $r(1)=1$ , retaining only the term $m=n=1$ and using the classical lower bound for Euler’s totient function, we obtain the crude lower bound

(3.2) $$ \begin{align} S_1 \ge q^{1-o(1)}. \end{align} $$

For the principal character $\chi _0$ , it follows from the prime number theorem that

$$ \begin{align*} |R(\chi_0)|^2 = 2^{2\pi(Y)} = q^{\eta\log 4+o(1)}. \end{align*} $$

Together with

$$ \begin{align*} |P(\chi_0)| \ll_{\sigma,\ell} \sum_{n \le q} \frac{(\log n)^\ell}{n^\sigma} \ll_{\sigma,\ell} q^{1-\sigma}(\log q )^\ell, \end{align*} $$

this yields

$$ \begin{align*} |P(\chi_0)||R(\chi_0)|^2 \ll_{\sigma,\ell} q^{1-\sigma+\eta\log 4+o(1)}. \end{align*} $$

It follows from $0 < \eta < (2\log 4)^{-1} \sigma $ that

$$ \begin{align*} 1-\sigma+\eta\log4 <1. \end{align*} $$

Therefore, the contribution of the principal character $\chi _0$ is negligible compared with the lower bound for $S_1$ given in (3.2).

Denote by $\mathcal {G}_q^\ast $ the set of all nonprincipal Dirichlet characters modulo q, that is, $\mathcal {G}_q^\ast = \mathcal {G}_q \setminus \{\chi _0\}$ and define the two sums

$$ \begin{align*} S_1^\ast := S_1^\ast(R,q) = \sum_{\chi \in \mathcal{G}_q^\ast}|R(\chi)|^2, \quad S_2^\ast := S_2^\ast(R,q) = \sum_{\chi \in \mathcal{G}_q^\ast}P(\chi) |R(\chi)|^2. \end{align*} $$

Combining $S_1^\ast \le S_1$ and (3.2) gives

$$ \begin{align*} \frac{S_2^\ast}{S_1^\ast} \ge \frac{S_2}{S_1} + O(q^{-\sigma+\eta\log 4+o(1)} ). \end{align*} $$

Thus,

(3.3) $$ \begin{align} \max_{\chi\in \mathcal{G}_q^\ast} |P(\chi)| \ge \frac{S_2}{S_1} + O(q^{-\sigma+\eta\log 4+o(1)} ). \end{align} $$

We now establish a lower bound for the ratio $S_2/S_1$ . To this end, expanding $P(\chi )$ and $|R(\chi )|^2$ in $S_2$ ,

$$ \begin{align*} S_2 &\, = \sum_{\chi \in \mathcal{G}_q}\sum_{k < q}\frac{(\log k )^\ell \chi(k)}{k^\sigma} \sum_{m,n\ge 1} r(m)r(n)\chi(m)\overline\chi(n) \\ &\, = \sum_{k < q}\frac{(\log k )^\ell}{k^\sigma}\sum_{m,n\ge 1} r(m)r(n) \sum_{\chi \in \mathcal{G}_q}\chi(k)\chi(m)\overline\chi(n). \end{align*} $$

The orthogonality of Dirichlet characters implies that

$$ \begin{align*} S_2 = \sum_{k < q}\frac{(\log k )^\ell}{k^\sigma} \phi(q) \sum_{\substack{m,n \ge 1 \\ mk \equiv n\pmod q}} r(m)r(n). \end{align*} $$

Since all terms on the right-hand side are nonnegative, we may retain only those with $k\mid n$ , which gives the crude lower bound

(3.4) $$ \begin{align} S_2 \ge \phi(q)\sum_{k < q}\frac{(\log k )^\ell}{k^\sigma}r(k)\sum_{\substack{m,n \ge 1 \\ mk \equiv nk \pmod q}} r(m)r(n). \end{align} $$

For every integer k occurring in (3.4), we have $1\leq k<q$ . Since q is prime, it follows that $(k,q)=1$ . Therefore,

$$ \begin{align*} mk\equiv nk\pmod q \quad\Longleftrightarrow\quad m\equiv n\pmod q. \end{align*} $$

Combining this observation with (3.1) and (3.4),

(3.5) $$ \begin{align} \frac{S_2}{S_1} \geq \sum_{k<q}\frac{(\log k)^\ell}{k^\sigma}r(k) = \sum_{k\leq q}\frac{(\log k)^\ell}{k^\sigma}r(k) =:Q_\ell(q). \end{align} $$

Here, the second equality follows from $r(q)=0$ since $Y<q$ for sufficiently large q.

If $k> q$ , then $k^{-\sigma } \le q^{-\sigma /2} k^{-\sigma /2}$ ; hence,

$$ \begin{align*} \sum_{k> q}\frac{(\log k )^\ell}{k^\sigma}r(k) \le q^{-\sigma/2} \sum_{k \in \mathbb{N}}\frac{(\log k )^\ell}{k^{\sigma/2}}r(k). \end{align*} $$

The sum on the right-hand side of the above formula is bounded by

$$ \begin{align*} \sum_{k=1}^{\infty}\frac{r(k)}{k^{\sigma/4}}=\prod_{p\leq Y}\bigg(1-\frac{1}{2p^{\sigma/4}}\bigg)^{-1}. \end{align*} $$

The prime number theorem gives

$$ \begin{align*} \log \prod_{p\leq Y}\bigg(1-\frac{1}{2p^{\sigma/4}}\bigg)^{-1}\ll_{\sigma}\sum_{p\leq Y}p^{-{\sigma}/{4} }\ll_\sigma\frac{Y^{1-\sigma/4}}{\log Y}=o(\log q). \end{align*} $$

Thus,

$$ \begin{align*} \sum_{k \in \mathbb{N}}\frac{(\log k )^\ell}{k^{\sigma/2}}r(k) \le q^{o(1)}. \end{align*} $$

The above argument shows that

(3.6) $$ \begin{align} Q_\ell(q) = \sum_{k \in \mathbb{N}}\frac{(\log k )^\ell}{k^\sigma}r(k) -o(1). \end{align} $$

Since $Y\ge 2$ for sufficiently large q, we have $r(2)=1/2$ . Thus,

$$ \begin{align*} \sum_{k\in\mathbb{N}}\frac{(\log k)^\ell}{k^\sigma}r(k) \geq\sum_{m\in\mathbb{N}}r(2m)\frac{(\log 2m)^\ell}{(2m)^\sigma}\geq\frac{(\log 2)^\ell}{2^{\sigma+1}}\sum_{m\in\mathbb{N}}\frac{r(m)}{m^\sigma}. \end{align*} $$

Here, we employ the fact that $r(n)$ is a completely multiplicative function. Combining this with (3.6), we obtain the lower bound

(3.7) $$ \begin{align} Q_\ell(q) \gg \sum_{k \in \mathbb{N}} \frac{r(k)}{k^\sigma}. \end{align} $$

Set

$$ \begin{align*} F(\sigma) := \sum_{k \in \mathbb{N}} \frac{r(k)}{k^\sigma}. \end{align*} $$

Then, by the definition of $r(n)$ ,

$$ \begin{align*} F(\sigma)=\prod_{p\leq Y}\bigg(1-\frac{1}{2p^\sigma}\bigg)^{-1}. \end{align*} $$

Thus,

$$ \begin{align*} \log F(\sigma)=-\sum_{p\leq Y}\log\bigg(1-\frac{1}{2p^\sigma}\bigg) =\sum_{p\leq Y}\frac{1}{2p^\sigma}+O\bigg(\sum_{p\leq Y}\frac{1}{p^{2\sigma}}\bigg) =\frac 12\sum_{p\leq Y}\frac{1}{p^\sigma}+O_\sigma(1). \end{align*} $$

The prime number theorem shows that, when $1/2<\sigma <1$ ,

$$ \begin{align*} \sum_{p \le Y} \frac{1}{p^\sigma}= \bigg(\frac{1}{1-\sigma}+o(1)\bigg) \frac{Y^{1-\sigma}}{\log Y}. \end{align*} $$

Recalling that $Y= \eta \log q \log \log q$ ,

(3.8) $$ \begin{align} \log F(\sigma) = \bigg(\frac{\eta^{1-\sigma}}{2(1-\sigma)}+o(1)\bigg) \frac{(\log q)^{1-\sigma}}{(\log\log q)^{\sigma}}. \end{align} $$

Combining (3.3), (3.5), (3.7) and (3.8), and using Lemma 2.1, we complete the proof of Theorem 1.1.