January 01, 1970
In this paper, we apply the long resonator method to exhibit large values at the central point of the family of quadratic Dirichlet \(L\)-functions of prime-related moduli under the generalized Riemann hypothesis.
Dirichlet \(L\)-functions, large values, resonance method, character sums
Much progress has been made towards understandings of maximum sizes of \(L\)-functions ever since the introduction of the resonance method by K. Soundararajan in Sound08?, who showed that for the Riemann zeta function \(\zeta(s)\) and sufficiently large \(T\), \[\max_{T\leq t\leq 2T}|\zeta( \frac{1}{2}+it)|\geq \exp\left((1+o(1))\sqrt{\frac{\log T}{\log_2 T}}\right),\] where we denote \(\log_j\) the \(j\)-fold iterated logarithm throughout the paper. The above result was improved by A. Bondarenko and K. Seip in BS18? using the long resonators introduced by C. Aistleitner in Aistleitner16? to be \[\max_{0\leq t\leq T}|\zeta(\frac{1}{2}+it)|\geq \exp\left((1+o(1))\sqrt{\frac{\log T \log_3 T}{\log_2 T}}\right).\] The constant \(1\) was further improved to \(\sqrt{2}\) by R. de la Bretèche and G. Tenenbaum in BT19?.
It is also shown in Sound08? that for sufficiently large \(X\), \[\max_{\substack{d \text{ fundamental discriminant }\\X<|d|\leq 2X}}|L(\frac{1}{2}, \chi_d)|\geq \exp\left(\left(\frac{1}{\sqrt{5}}+o(1)\right)\sqrt{\frac{\log X }{\log_2 X}}\right),\] where \(\chi_d=\left(\frac{d}{\cdot}\right)\) is the Kronecker symbol.
In DM25?, P. Darbar and G. Maiti employed the long resonator technique of Bondarenko–Seip to study the large value of \(L(1/2, \chi_d)\) to show that under the Generalized Riemann Hypothesis (GRH), for sufficiently large \(X\), \[\begin{align} \label{d} \max_{\substack{d \text{ fundamental discrimint }\\X<|d|\le 2X}}|L(\frac{1}{2}, \chi_{d})|\geq \exp\left(\left(\frac{1}{2} +o(1) \right)\sqrt{\frac{\log X \log_3 X}{\log_2 X}}\right). \end{align}\tag{1}\]
Throughout the paper, we reserve the letters \(p, q\) for primes. In FHX26?, M. Fan, S. Hua and S. Xie proved that for sufficiently large \(X\), \[\begin{align} \label{p} \max_{\substack{X< p \leq 2X \\ p \equiv 1 \negthickspace \negthickspace \negthickspace \pmod8}}|L(\frac{1}{2}, \chi_p)|\geq \exp\left(\left(\sqrt{\frac{8}{45}}+o(1)\right)\sqrt{\frac{\log X }{\log_2 X}}\right). \end{align}\tag{2}\]
It is the aim of this paper to carry the approach of Darbar and Maiti to study large values at the central point of the family of quadratic Dirichlet \(L\)-functions of prime moduli under GRH. Due to some technical reasons, we consider the family of \(L\)-functions \(\{ L(s, \chi_{8q}) \}\) with \(q\) running over odd primes. Here we note as shown in MVa1?, the character \(\chi_{8q}\) is primitive modulo \(8q\).
Our main result investigates the maximum sizes of \(L(s, \chi_{8q})\) for \(q\) running over primes.
Theorem 1. With the notation as above and assuming the truth of GRH. We have for sufficiently large \(X\), \[\begin{align} \label{Llowerbound} \begin{aligned} \max_{\substack{q \text{ \rm prime }\\X< q \le 2X}}\Big|L(\frac{1}{2}, \chi_{8q})\Big |\geq \exp\left(\left(\frac{1}{2} +o(1) \right)\sqrt{\frac{\log X \log_3 X}{\log_2 X}}\right). \end{aligned} \end{align}\qquad{(1)}\]
As mentioned above, our proof of Theorem 1 is based on the approaches introduced in DM25?. The result of Theorem 1 implies that given in 1 . Without much effort, one may show that Theorem 1 is also valid with \(L(\frac{1}{2}, \chi_{8q})\) replaced by \(L(\frac{1}{2}, \chi_{q})\) for \(q\) being a fundamental discriminant so that our result also improves that given in 2 under GRH.
We have the following approximate functional equation concerning \(L(1/2, \chi_{8q})\) given in sound1?.
Lemma 1. For any odd prime \(q\), we have \[\begin{align} \begin{aligned} L\left( \frac{1}{2}, \chi_{8q} \right) = & 2\sum^{\infty}_{\substack{n=1}} \frac{\chi_{8q}(n)}{\sqrt{n}} V \left(\frac{ n}{\sqrt{q}} \right), \end{aligned} \end{align}\] where for any real number \(x>0\), \[\begin{align} V(x) = \frac{1}{2 \pi i} \int\limits\limits_{(2)} \left(\frac{8}{\pi}\right)^{s/2} \frac{\Gamma(s/2+1/4)}{\Gamma(1/4)} x^{-s} \frac{\mathrm{d}s}{s}. \end{align}\]
As shown in sound1?, the function \(V(x)\) is real-valued, smooth on \((0, +\infty)\), bounded as \(x\) approaches \(0\) and decays exponentially as \(x\to +\infty\). More precisely, we have for any \(\varepsilon>0\), \[\label{24607} V\left (x \right) = 1+O(x^{1/2-\varepsilon}) \; for \; 0<x <1 \quad and \quad V^{(j)}\left (x \right) =O(e^{-x}) \; for \; x >0, \; j \geq 0.\tag{3}\]
We also note that \[\begin{align} \label{eq:Vder} V'(x) = -\frac{1}{2 \pi i} \int\limits\limits_{(2)} \left(\frac{8}{\pi}\right)^{s/2} \frac{\Gamma(s/2+1/4)}{\Gamma(1/4)} x^{-s-1} ds. \end{align}\tag{4}\] Recall that for \(\Re(s)>0\), \[\begin{align} \Gamma(s)=\int^{\infty}_0e^{-x}x^{s-1}dx. \end{align}\] It follows from the inverse Mellin transformation that for \(x>0\), \[\begin{align} e^{-x}=\frac{1}{2 \pi i} \int\limits\limits_{(2)}\Gamma(s)x^{-s}ds. \end{align}\] We deduce from the above that \[\begin{align} 2x^{1/4}e^{-x}=\frac{1}{2 \pi i} \int\limits\limits_{(2)}\Gamma(\frac{s}{2}+\frac{1}{4})x^{-s}ds. \end{align}\] It follows from 4 and the above that we have \(V'(x)<0\) for \(x>0\). Note moreover that by 3 we have \(\lim_{x \rightarrow \infty}V(x)=0\). Thus we conclude that \(V(x)\) is positive for \(x \geq 0\).
We define \(\delta_{c=\square}=1\) if \(c\) is a perfect square and \(\delta_{c=\square}=0\) otherwise. In the remainder of the paper, let \(\Phi\) denote a smooth, non-negative function compactly supported on \([1,2]\) satisfying \(\Phi(x) =1\) for \(x\in [5/4,7/4]\). The Mellin transform of \(\Phi(x)\) is wirtten as \({\widehat \Phi}(s)\) so that for any complex number \(s\), \[{\widehat \Phi}(s) = \int\limits_{0}^{\infty} \Phi(x)x^{s}\frac{\mathrm{d}x}{x}.\] We have the following result on the smoothed quadratic character sums.
Lemma 1. With the notation as above and assuming the truth of GRH. Let \(c\) be a positive odd integer and \(\Phi(X)\) be a smooth function fitting the above descriptions. Then for any \(\varepsilon>0\), \[\sum_{(q,2)=1} (\log q) \chi_{8q}(c) \Phi \left( \frac{q}{X} \right) = \delta_{c=\square}\widehat{\Phi}(1)X+O \left( X^{1/2+\varepsilon}\log (c+2) \right).\]
Let \(a\in (1, \infty), \delta\in (0,1)\) be fixed such that \(1<a<\frac{1}{\delta}\), and let \(N\) be a large number to be specified later. Let \([x]\) denote the largest integer not exceeding \(x\) for any real \(x\). For \(k=1,\ldots,[(\log_{2}N)^{\delta}]\), denote \(\mathcal{P}_{k}\) the set of all primes \(p\) such that \[e^{k}\log N \log_{2} N < p \le e^{k+1}\log N \log_{2} N.\] We further denote \(\mathcal{R}_{k}\) the set of positive integers that have at least \(\frac{a\log N}{k^{2}\log_{3}N}\) prime divisors in \(\mathcal{P}_{k}\), and \(\mathcal{R}^{\prime}_{k}\) the set of positive integers from \(\mathcal{R}_{k}\) that have all their prime divisors in \(\mathcal{P}_{k}\).
We let \(\mathcal{P}\) be the union of \(\mathcal{P}_{k}\) so that it is the set of all primes \(p\) such that \[e\log N \log_{2} N< p \le e^{(\log_{2} N)^{\delta}} \log N \log_{2} N.\] We define a multiplicative function \(\psi\) supported on the set of square-free integers such that for any \(p \in\mathcal{P}\), \[\begin{align} \label{psidef} \psi(p)=\sqrt{\frac{\log N \log_{2}N }{\log_{3}N}}p^{-1/2}\left(\log p-\log(\log N \log_{2}N)\right)^{-1}. \end{align}\tag{5}\] We further define \(\psi(p)=0\) for \(p \notin \mathcal{P}\).
We now set \[\mathcal{R}:=\rm{supp}(\psi)\setminus\displaystyle \bigcup_{k=1}^{[(\log_{2}N)^{\delta}]}\mathcal{R}_{k}.\] Then, we define the resonator for \(L(\frac{1}{2}, \chi_{8q})\) for any prime \(q\) to be the Dirichlet polynomial \[R_q:= \sum_{m\in \mathcal{R}}\psi(m)\chi_{8q}(m).\]
The following lemma, taken from DM25?, gives estimations on the sizes of \(|\mathcal{R}|\), respectively.
Lemma 1. With the notation as above. Suppose that \(1<a<\frac{1}{\delta}\). Then we have \(|\mathcal{R}|\leq N\) for \(N\) large enough.
We now define \[\begin{align} \mathcal{A}_{N}:=\frac{1}{\displaystyle \sum_{m \in \mathbb{N}} \psi(m)^{2} }\sum_{n\in \mathbb{N}}\frac{\psi{(n)}}{\sqrt{n}}\sum_{\substack{l|n }}\psi(l)\sqrt{l}. \end{align}\]
We end this section by including three results from BS18? concerning various sums related to \(\mathcal{A}_{N}\).
Lemma 1. With the notation as above. We have as \(N\to\infty\), \[\mathcal{A}_{N}\ge \exp\left((\delta + o(1))\sqrt{\frac{\log N\log_{3}N}{\log_{2}N}}\right).\]
Lemma 1. With the notation as above. We have as \(N\to\infty\), \[\begin{align} \frac{1}{\displaystyle\sum_{m\in \mathbb{N}} \psi(m)^{2}}\sum_{\substack{n\in \mathbb{N}\\ n\notin\mathcal{R}}}\frac{\psi{(n)}}{\sqrt{n}}\sum_{\substack{l|n }}\psi(l)\sqrt{l}=o(\mathcal{A}_{N}). \end{align}\]
Lemma 1. With the notation as above. We have as \(N\to\infty\), for any \(\varepsilon>0\), \[\frac{1}{\displaystyle \sum_{m \in \mathbb{N}} \psi(m)^{2}}\sum_{\substack{n\in \mathcal{R} }}\frac{\psi{(n)}}{\sqrt{n}}\sum_{\substack{l|n\\ l \le n/N^{\varepsilon}}}\psi(l)\sqrt{l}=o(\mathcal{A}_{N}),\] where the implicit constant only depends on \(\varepsilon\).
Let \(\Phi\) be the function described in Section 2.2. We set \[\begin{align} \mathcal{S}_{1}:=\sideset{}{^*}\sum_q (\log q)L(\frac{1}{2},\chi_{8q}) R_q^2 \Phi (\frac{q}{X}), \quad \mathcal{S}_{2}:=\sideset{}{^*}\sum_{q }(\log q)R_q^2\Phi (\frac{q}{X}). \end{align}\] As \(R^2_d \geq 0\) and \(L(\frac{1}{2},\chi_{8q}) \in \mathbb{R}\), we then observe that \[\begin{align} \label{maxlower} \max_{\substack{X< q \le 2X}}\Big|L(\frac{1}{2}, \chi_{8q})\Big |\ge \frac{\mathcal{S}_1}{\mathcal{S}_2}. \end{align}\tag{6}\] It remains to establish a lower bound for \(\mathcal{S}_{1}\) and an upper bound for \(\mathcal{S}_{1}\). We first note that \[\begin{align} \label{S1eval} \begin{aligned} \mathcal{S}_{1}=&\sideset{}{^*}\sum_q (\log q)L(\frac{1}{2},\chi_{8q}) R_q^2 \Phi (\frac{q}{X}) \\ =&2\sum_{m, n\in \mathcal{R}} \psi(m) \psi(n)\sum_{l\geq 1 } \frac{1}{\sqrt{l}}\sideset{}{^*}\sum_{q}\chi_{8q}(lmn) V\left(\frac{l}{\sqrt{q}} \right). \end{aligned} \end{align}\tag{7}\] We now apply Lemma 1 and partial summation to see that \[\begin{align} \begin{aligned} & \sideset{}{^*}\sum_{q}\chi_{8q}(lmn) V\left(\frac{l}{\sqrt{q}} \right) \Phi (\frac{q}{X}) \\ =& \int^{2X}_X V\left(\frac{l}{\sqrt{t}} \right) d \left( t\delta_{lmn=\square}{\widehat \Phi}(1)+ O\left(t^{1/2+\varepsilon}\log (lmn+2) \right) \right) \\ =& {\widehat \Phi}(1)X \delta_{lmn=\square}\int^{2}_1 V\left(\frac{l}{\sqrt{Xt}} \right) dt \\ &+V\left(\frac{l}{\sqrt{t}} \right)O\left(t^{1/2+\varepsilon}\log (lmn+2) \right)\Big |^{2X}_X -\int^{2X}_XO\left(t^{1/2+\varepsilon}\log (lmn+2) \right)V'\left(\frac{l}{\sqrt{t}}\right ) \frac{l}{2t^{3/2}}dt \\ =: & {\widehat \Phi}(1)X \delta_{lmn=\square}\int^{2}_1 V\left(\frac{l}{\sqrt{Xt}} \right) dt+R. \end{aligned} \end{align}\] It follows from this and 7 that \[\begin{align} \mathcal{S}_{1}={\widehat \Phi}(1)X \sum_{m, n \in \mathcal{R}}&\psi(m)\psi(n)\sum_{\substack{lmn =\square}} \frac{1}{\sqrt{l}}\int_{1}^{2}V\left(\frac{l}{\sqrt{Xt}} \right) dt+O\Bigg(\sum_{m, n\in \mathcal{R}} \psi(m) \psi(n)\sum_{l\geq 1 } \frac{1}{\sqrt{l}}R\Bigg). \end{align}\] In view of the rapid decay of \(V\) and \(V'\) given in 3 , we see that we may restrict the sum over \(l\) to be \(l \leq X^{1/2+\varepsilon}\) for any \(\varepsilon>0\) in the error term above. We further set \(N= X^{\frac{1}{4}-5\varepsilon}\) for some \(0<\varepsilon<1/20\) and observe that \(|\mathcal{R}|\leq N\) from Lemma 1. It follows that \[\begin{align} R \ll X^{1/2+\varepsilon}. \end{align}\] We deduce from the above that \[\begin{align} \label{S1exp} \begin{aligned} \mathcal{S}_{1}=& {\widehat \Phi}(1)X\sum_{m, n \in \mathcal{R}}\psi(m)\psi(n)\sum_{\substack{ lmn =\square}} \frac{1}{\sqrt{l}}\int_{1}^{2}V\left(\frac{N(l)}{\sqrt{Xt}} \right) dt+O\Bigg(X^{1/2 +\varepsilon}\sum_{\substack{l \leq X^{1/2+\varepsilon}}}\frac{1}{\sqrt{l}} \left(\sum_{m\in \mathcal{R}}\psi(m)\right)^2\Bigg) \\ =& {\widehat \Phi}(1)X\sum_{m, n \in \mathcal{R}}\psi(m)\psi(n)\sum_{\substack{ lmn =\square}} \frac{1}{\sqrt{l}}\int_{1}^{2}V\left(\frac{N(l)}{\sqrt{Xt}} \right) dt+O\Bigg(X^{3/4 +2\varepsilon}\left(\sum_{m\in \mathcal{R}}\psi(m)\right)^2\Bigg) \\ =& {\widehat \Phi}(1)X\sum_{m, n \in \mathcal{R}}\psi(m)\psi(n)\sum_{\substack{ lmn =\square}} \frac{1}{\sqrt{l}}\int_{1}^{2}V\left(\frac{N(l)}{\sqrt{Xt}} \right) dt+O\Bigg(X^{3/4 +2\varepsilon}|\mathcal{R}|\sum_{m\in \mathcal{R}}\psi^2(m)\Bigg), \end{aligned} \end{align}\tag{8}\] where the last estimation above follows from the Cauchy–Schwarz inequality.
As \(\psi(n) \geq 0\) and \(V(x) \geq 0\), we may thus keep only the case \(lm=n\) in the main term of 8 . Together with the observation that \(|\mathcal{R}|\leq N\) from Lemma 1, we see that \[\begin{align} \label{S1remainder} \mathcal{S}_{1} \ge & {\widehat \Phi}(1)X\sum_{n \in\mathcal{R}} \frac{\psi(n)}{\sqrt{n}} \sum_{\substack{m \mid n }} \psi(m)\sqrt{m}\int_{1}^{2}V\left(\frac{n}{m\sqrt{Xt}} \right) dt+O\left(X^{3/4+2\varepsilon}N\left(\sum_{\substack{m\in \mathcal{R}}}\psi(m)^2\right)\right). \end{align}\tag{9}\]
As \(N= X^{\frac{1}{4}-5\varepsilon}\), it follows that when \(m\ge n/N^{\varepsilon}\), we have \(0< \frac{n}{m\sqrt{Xt}}<1\) for \(1 \leq t \leq 2\). We now apply the estimation \(V(x)=1 + O\left(x^{\frac{1}{2}-\varepsilon}\right)\) given in 3 to deduce from the above that \[\begin{align} \label{S1lower} \begin{aligned} \mathcal{S}_{1} \ge & {\widehat \Phi}(1)X\sum_{n \in\mathcal{R}} \frac{\psi(n)}{\sqrt{n}} \sum_{\substack{m\mid n\\ m\ge n/N^{\varepsilon}}} \psi(m)\sqrt{m} \\ & +O\left(X^{3/4+2\varepsilon}\sum_{n \in\mathcal{R}} \psi(n)\sum_{\substack{m\mid n\\ m\ge n/N^{\varepsilon}}} \psi(m) \right)+O\left(X^{1-\varepsilon}\left(\sum_{\substack{m\in \mathcal{R}}}\psi(m)^{2}\right)\right). \end{aligned} \end{align}\tag{10}\]
We observe from 5 that we have \(0< \psi(p) \leq 1\) for \(p \in \mathcal{P}\). It follows from this that \[\begin{align} \sum_{n \in\mathcal{R}} \psi(n)\sum_{\substack{m\mid n\\ m\ge n/N^{\varepsilon}}} \psi(m) \ll \sum_{n \in\mathcal{R}} \psi(n)\sum_{\substack{ m\mid n}} \psi(m)=\sum_{m \in\mathcal{R}} \psi(m)\sum_{\substack{n \in\mathcal{R} \\ m\mid n}}\psi(n) \ll |\mathcal{R}|\sum_{m \in\mathcal{R}} \psi^2(m). \end{align}\]
As \(|\mathcal{R}|\leq N\) from Lemma 1 and \(N= X^{\frac{1}{4}-5\varepsilon}\), we deduce from the above and 10 that \[\begin{align} \begin{aligned} \mathcal{S}_{1} \ge & {\widehat \Phi}(1)X\sum_{n \in\mathcal{R}} \frac{\psi(n)}{\sqrt{n}} \sum_{\substack{m\mid n\\ m\ge n/N^{\varepsilon}}} \psi(m)\sqrt{m}+O\left(X^{1-\varepsilon}\left(\sum_{\substack{m\in \mathcal{R}}}\psi(m)^{2}\right)\right). \end{aligned} \end{align}\]
We now apply above together with Lemma 1, Lemma 1, and Lemma 1 to see that \[\begin{align} \label{S1lower2} \begin{aligned} \mathcal{S}_{1}\ge& \left(1+o(1)\right){\widehat \Phi}(1)X\exp\left((\delta+ o(1))\sqrt{\frac{\log N\log_{3}N}{\log_{2}N}}\right)\left(\sum_{\substack{m\in \mathcal{R}}}\psi(m)^{2}\right) +O\left(X^{1-\varepsilon}\left(\sum_{\substack{m\in \mathcal{R}}}\psi(m)^{2}\right)\right)\\ \ge &\left(1+o(1)\right){\widehat \Phi}(1)X \exp\left(\left(\delta \sqrt{\frac{1}{4}-5\varepsilon}+o(1)\right)\sqrt{\frac{\log X\log_{3} X}{\log_{2} X}}\right) \left(\sum_{\substack{m\in \mathcal{R}}}\psi(m)^{2}\right). \end{aligned} \end{align}\tag{11}\]
Next, we proceed to establish an upper bound of \(\mathcal{S}_2\). Again, applying Lemma 1 and arguing similar to those that lead to 9 imply that \[\begin{align} \mathcal{S}_{2}=& \sideset{}{^*}\sum_{q }(\log q)R_q^2\Phi (\frac{q}{X})=\sum_{m, n \in \mathcal{R}}\psi(m) \psi(n) \sideset{}{^*}\sum_{q }(\log q)\chi_{8q}(m n)\Phi (\frac{q}{X})\\ =& {\widehat \Phi}(1)X \sum_{\substack{m, n \in \mathcal{R} \\ m n =\square}}\psi(m) \psi(n) +O\Bigg(X^{\frac{1}{2} + \varepsilon} \sum_{\substack{m,n \in \mathcal{R}}}\psi(m) \psi(n)\log (mn+2)\Bigg)\\ =& {\widehat \Phi}(1)X \sum_{\substack{n \in \mathcal{R}}}\psi(n)^{2} + O\left(X^{1/2 +\varepsilon} N \sum_{\substack{m\in \mathcal{R}}} \psi(m)^{2} \right), \end{align}\] where the last equality above follows by noting that \(mn=\square\) implies that \(m=n\) since \(m\) and \(n\) are square-free.
Again using \(|\mathcal{R}|\leq N= X^{\frac{1}{4}-5\varepsilon}\), we deduce from the above that \[\begin{align} \mathcal{S}_{2} \le \left(1+o(1)\right){\widehat \Phi}(1)X \left(\sum_{\substack{m\in \mathcal{R}}}\psi(m)^{2}\right). \end{align}\] Hence, for sufficiently large \(X\) and arbitrary small \(\varepsilon>0\), we deduce from 6 , 11 and the above that \[\begin{align} \max_{\substack{q \text{ \rm prime }\\X< q \le 2X}}\Big|L(\frac{1}{2}, \chi_{8q})\Big | &\geq \frac{\mathcal{S}_1}{\mathcal{S}_2}\geq \exp\left(\left(\delta \sqrt{\frac{1}{4}-5\varepsilon}+o(1)\right)\sqrt{\frac{\log X\log_{3} X}{\log_{2} X}}\right). \end{align}\] By letting \(\delta \rightarrow 1^+\), we obtain the desired estimation given in ?? . This completes the proof of Theorem 1.
P. G. is supported in part by NSFC grant 12471003.