Lower bounds for negative moments of Dirichlet $L$ -functions to a fixed modulus

Peng Gao School of Mathematical Sciences, Beihang University, Beijing 100191, China [email protected]

Abstract.

We establish lower bounds for the $2k$ -th moment of central values of the family of primitive Dirichlet $L$ -functions to a fixed prime modulus for all real $k<0$ , assuming the non-vanishing of these $L$ -values.

Mathematics Subject Classification (2020): 11M06

Keywords: lower bounds, negative moments, Dirichlet $L$ -functions

1. Introduction

The generalized Riemann hypothesis (GRH) asserts that all non-trivial zeros of Dirichlet $L$ -functions can be written as $\rho=1/2+i\gamma$ with $\gamma\in\mathbb{R}$ . Moreover, it is believed that there are no $\mathbb{Q}$ -linear relations among the non-negative $\gamma$ ’s. In particular, this implies that $L(\tfrac{1}{2},\chi)\neq 0$ for all primitive Dirichlet characters $\chi$ and it is known as a conjecture of S. Chowla [chow] when $\chi$ is quadratic.

One way to investigate the non-vanishing problem is to evaluate the one-level densities of low-lying zeros of families of $L$ -functions. In fact, the density conjecture of N. Katz and P. Sarnak [KS1, K&S] implies that $L(1/2,\chi)\neq 0$ for almost all primitive Dirichlet $L$ -functions. Computing essentially the one-level density of low-lying zeros of the corresponding families of Dirichlet $L$ -functions for test functions whose Fourier transforms being supported in $(-2,2)$ , M. R. Murty [Murty] showed that under GRH, at least $50\%$ of the primitive Dirichlet $L$ -functions do not vanish at the central point. See also the work of H. P. Hughes and Z. Rudnick In [HuRu] regarding the one-level density of low-lying zeros of the family of primitive Dirichlet $L$ -functions to a fixed prime modulus.

Another way to address whether $L(\tfrac{1}{2},\chi)=0$ or not is to study moments of families of $L$ -functions. In [BM], B. Balasubramanian and V. K. Murty showed that $L(1/2,\chi)\neq 0$ for at least $4\%$ of Dirichlet characters $\chi$ to a fixed modulus $q$ by evaluating the first and second mollified moments of $L(1/2,\chi)$ . For a prime $q$ , this proportion was improved to $1/3$ by H. Iwaniec and P. Sarnak [I&S], to $34.11\%$ by H. M. Bui [Bui], to $3/8$ by R. Khan and H. T. Ngo [KN], and to $5/13$ by R. Khan, D. Milićević and H. T. Ngo [KMN22].

Because of the important role played by moments of $L$ -functions concerning the non-vanishing issue, a considerable amount of work has been done in this direction. For the family of Dirichlet $L$ -functions to a fixed modulus $q\not\equiv 2\pmod{4}$ (to ensure primitive Dirichlet characters modulo $q$ exist), it is widely believed that (see [R&Sound]) for all real $k\geq 0$ ,

(1.1)

\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2k}\sim C_{k}\phi^{*}(q)(\log q)^{k^{2}},

where the numbers $C_{k}$ are explicit constants, $\phi^{*}(q)$ denotes the number of primitive characters modulo $q$ and where we denote $\sideset{}{{}^{*}}{\sum}$ the sum over primitive Dirichlet characters modulo $q$ throughout the paper.

The formula given in (1.1) was conjectured by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith in [CFKRS] for all positive integral values of $k$ and is well-known for $k=1$ . The case $k=2$ was established by D. R. Heath-Brown [HB81] for almost all $q$ and was later shown to hold for all $q$ by K. Soundararajan [Sound2007]. Subsequent improvements on the error terms in Soundararajan’s result can be found in [Young2011, BFKMM1, BFKMM, Wu2020, BPRZ].

Although it is challenging to prove (1.1) even for integers $k\geq 3$ , much progress has been made for building upper and lower bounds of the conjectured order of magnitude for the expression on the left-hand side of (1.1). In fact, there are now several systematic approaches towards establishing sharp lower and upper bounds. Notably, there are the upper bounds principle due to M. Radziwiłł and K. Soundararajan [Radziwill&Sound] as well as the lower bounds principle due to W. Heap and K. Soundararajan [H&Sound]. Other methods can be found in [Sound01, HB2010, Harper, R&Sound1, Radziwill&Sound1, C&L, Gao2024-6]. These results together imply that for large prime $q$ and any real number $k\geq 0$ , we have

(1.2)

\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2k}\asymp\phi^{*}(q)(\log q)^{k^{2}}.

Here we note that one needs to assume GRH in order to show that $\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2k}\ll_{k}\phi^{*}(q)(\log q)^{k^{2}}$ for $k>1$ .

As the order of magnitude for the non-negative moments of the family of Dirichlet $L$ -functions to a fixed prime modulus are established in (1.2), it is natural to turn to the negative moments of this family and we assume that $L(\tfrac{1}{2},\chi)\neq 0$ for each $\chi$ in the family in order for negative powers of $|L(\tfrac{1}{2},\chi)|$ to be meaningful. An analogue case for the family of quadratic Dirichlet $L$ -functions has been considered by the author in [Gao2022-1]. As already being pointed out in [Gao2022-1], the behaviour of the negative moments may be more difficult to predict compared to that of the positive ones. One can also see this from the case $\beta=2$ of [FK, Corollary 1], where computations by P. J. Forrester and J. P. Keating based on random matrix theory suggests certain phase changes in the asymptotic formulas for the $2k$ -th moment of the family of Dirichlet $L$ –functions to a fixed modulus when $2k=-(2j-1)$ for any positive integer $j$ .

In this paper, we establish lower bounds for negative moments of the family of primitive Dirichlet $L$ -functions to a fixed prime modulus. Our result is as follows.

Theorem 1.1.

Let $q$ be a large prime number and assume that $L(\tfrac{1}{2},\chi)\neq 0$ for any primitive Dirichlet character $\chi\pmod{q}$ . Then we have for any real number $k<0$ ,

\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2k}\gg_{k}\phi^{*}(q)(\log q)^{k^{2}}.

We note that if one interprets $0^{2k}=+\infty$ for $k<0$ , then the statement of Theorem 1.1 is still valid without the assumption that $L(\tfrac{1}{2},\chi)\neq 0$ . We shall derive Theorem 1.1 by applying a variant of the lower bounds principle of W. Heap and K. Soundararajan [H&Sound]. Such an approach has already been employed by W. Heap, J. Li and J. Zhao [HLZ], as well as by P. Gao and L. Zhao [G&Zhao2023] to study lower bounds of discrete negative moments of the derivative of the Riemann zeta function $\zeta(s)$ at non-trivial zeros. We expect that the bounds given in Theorem 1.1 are sharp for $-1<2k<0$ , by taking account of the prediction in [FK] based on random matrix theory .

2. Proof of Theorem 1.1

2.1. Setup

For any real number $\ell$ , we let $\lceil\ell\rceil=\min\{m\in\mathbb{Z}:\ell\leq m\}$ be the celing function of $\ell$ . We define a sequence of even natural numbers $\{\ell_{j}\}_{1\leq j\leq R}$ such that $\ell_{1}=2\lceil N\log\log q\rceil$ and $\ell_{j+1}=2\lceil N\log\ell_{j}\rceil$ for $j\geq 1$ , where $N,M$ are two large natural numbers depending on $k$ only, and where $R$ is defined to be the largest natural number satisfying $\ell_{R}>10^{M}$ . We choose $M$ large enough to ensure that $\ell_{j}>\ell_{j+1}^{2}$ for all $1\leq j\leq R-1$ . It follows from this that

(2.1)

\displaystyle\sum^{R}_{j=1}\frac{1}{\ell_{j}}\leq\frac{2}{\ell_{R}}.

Let ${P}_{1}$ be the set of odd primes not exceeding $q^{1/\ell_{1}^{2}}$ and let ${P_{j}}$ be the set of primes lying in the interval $(q^{1/\ell_{j-1}^{2}},q^{1/\ell_{j}^{2}}]$ for $2\leq j\leq R$ . We write for each $1\leq j\leq R$ ,

{\mathcal{P}}_{j}(\chi)=\sum_{p\in P_{j}}\frac{1}{\sqrt{p}}\chi(p),\quad{\mathcal{Q}}_{j}(\chi,k)=\Big{(}\frac{12(1+|k|){\mathcal{P}}_{j}(\chi)}{\ell_{j}}\Big{)}^{(2-k/(1-k))\ell_{j}}.

We further define ${\mathcal{Q}}_{R+1}(\chi,k)=1$ .

We define for any non-negative integer $\ell$ and any complex number $x$ ,

E_{\ell}(x)=\sum_{j=0}^{\ell}\frac{x^{j}}{j!}.

Further, we define for each $1\leq j\leq R$ and any real number $\alpha$ ,

(2.2)

\displaystyle{\mathcal{N}}_{j}(\chi,\alpha)=E_{\ell_{j}}(\alpha{\mathcal{P}}_{j}(\chi)),\quad\mathcal{N}(\chi,\alpha)=\prod_{j=1}^{R}{\mathcal{N}}_{j}(\chi,\alpha).

Similarly, we define for each $1\leq j\leq R$ and any real number $\alpha$ ,

(2.3)

\displaystyle{\mathcal{M}}_{j}(\chi,\alpha)=E_{\ell_{j}}(\alpha\Re{\mathcal{P}}_{j}(\chi)),\quad\mathcal{M}(\chi,\alpha)=\prod_{j=1}^{R}{\mathcal{M}}_{j}(\chi,\alpha).

We now present a variant in our setting of the lower bounds principle of W. Heap and K. Soundararajan [H&Sound]. To do so, we first note that as $\ell_{j}$ is even for each $j$ , it follows from [Radziwill&Sound, Lemma 1] that ${\mathcal{M}}_{j}(\chi,\alpha)>0$ for any real $\alpha$ , which then implies that $\mathcal{M}(\chi,\alpha)>0$ as well. Moreover, it follows from [Gao2021-2, Lemma 4.1] that for any real number $\alpha$ ,

\displaystyle\mathcal{M}(\chi,\alpha)\mathcal{M}(\chi,-\alpha)\geq 1.

We apply the above to see for any $c>0$ ,

(2.4)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{|}\mathcal{N}(\chi,k)\Big{|}^{2}\leq&\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{|}\mathcal{N}(\chi,k)\Big{|}^{2}\Big{(}\mathcal{M}(\chi,k-1)\mathcal{M}(\chi,1-k)\Big{)}^{c}\\ =&\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}L(\frac{1}{2},\chi)^{-c}\cdot\big{(}L(\frac{1}{2},\chi)\mathcal{M}(\chi,k-1)\big{)}^{c}\cdot\Big{|}\mathcal{N}(\chi,k)\Big{|}^{2}\mathcal{M}(\chi,1-k)^{c}.\end{split}

As $k<0$ , we fix a constant $0<c<-2k/(1-k)$ satisfying

(2.5)

\displaystyle\begin{split}&0<\frac{c}{2}-\frac{c}{2k}<1.\end{split}

We then apply Hölder’s inequality with exponents $-2k/c,2/c,(1+(1-k)c/(2k))^{-1}$ to the right-hand side expression in (2.4) to see that

(2.6)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{|}\mathcal{N}(\chi,k)\Big{|}^{2}\leq&\Big{(}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2k}\Big{)}^{-c/(2k)}\Big{(}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2}|\mathcal{M}(\chi,k-1)|^{2}\Big{)}^{c/2}\\ &\times\Big{(}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{|}\mathcal{N}(\chi,k)^{2}\mathcal{M}(\chi,1-k)^{c}\Big{|}^{(1+(1-k)c/(2k))^{-1}}\Big{)}^{1+(1-k)c/(2k)}.\end{split}

In particular, we set $c=-k/(1-k)$ to see that the condition (2.5) is satisfied. It then follows from (2.6) that we have

(2.7)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{|}\mathcal{N}(\chi,k)\Big{|}^{2}\leq&\Big{(}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2k}\Big{)}^{1/(2(1-k))}\Big{(}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2}|\mathcal{M}(\chi,k-1)|^{2}\Big{)}^{-k/(2(1-k))}\\ &\times\Big{(}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{|}\mathcal{N}(\chi,k)^{2}\mathcal{M}(\chi,1-k)^{-k/(1-k)}\Big{|}^{2}\Big{)}^{1/2}.\end{split}

We deduce from (2.7) that in order to prove Theorem 1.1, it suffices to establish the following three propositions.

Proposition 2.2.

With the notation as above, we have

\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\big{|}\mathcal{N}(\chi,k)\big{|}^{2}\gg_{k}\phi^{*}(q)(\log q)^{k^{2}}.

Proposition 2.3.

With the notation as above, we have

\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{|}\mathcal{N}(\chi,k)^{2}\mathcal{M}(\chi,1-k)^{-k/(1-k)}\Big{|}^{2}\ll_{k}\phi^{*}(q)(\log q)^{k^{2}}.

Proposition 2.4.

With the notation as above, we have

\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\big{|}L(\tfrac{1}{2},\chi){\mathcal{M}}(\chi,k-1)\big{|}^{2}\ll_{k}\phi^{*}(q)(\log q)^{k^{2}}.

We shall prove the above Propositions in the rest of the paper.

2.5. Proof of Proposition 2.2

We denote $w(n)$ the multiplicative function satisfying $w(p^{\alpha})=\alpha!$ for prime powers $p^{\alpha}$ . We also define functions $b_{j}(n),1\leq j\leq R$ such that $b_{j}(n)=1$ when $\Omega(n)\leq\ell_{j}$ and the primes dividing $n$ are all from the interval $P_{j}$ , where $\Omega(n)$ denotes the number of prime powers dividing $n$ . For other values of $n$ , we set $b_{j}(n)=0$ . Using these notations, we see that for any real number $\alpha$ ,

(2.8)

{\mathcal{N}}_{j}(\chi,\alpha)=\sum_{n_{j}}\frac{1}{\sqrt{n_{j}}}\frac{\alpha^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\chi(n_{j}),\quad 1\leq j\leq R.

Observe that each ${\mathcal{N}}_{j}(\chi,\alpha)$ is a short Dirichlet polynomial as $b_{j}(n_{j})=0$ unless $n_{j}\leq(q^{1/\ell_{j}^{2}})^{\ell_{j}}=q^{1/\ell_{j}}$ . This together with (2.1) implies that ${\mathcal{N}}(\chi,k)$ is also a short Dirichlet polynomial whose lengths does not exceed $q^{1/\ell_{1}+\ldots+1/\ell_{R}}<q^{2/10^{M}}$ . Also, we see from (2.8) that for each $\chi$ modulo $q$ , including the case with $\chi$ being the principal character $\chi_{0}$ modulo $q$ ,

\displaystyle|{\mathcal{N}}(\chi,k)|^{2}\ll q^{2(1/\ell_{1}+\ldots+1/\ell_{R})}<q^{4/10^{M}}.

It follows from this that we have

(2.9)

\displaystyle\begin{split}&\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|{\mathcal{N}}(\chi,k)|^{2}\geq\sum_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|{\mathcal{N}}(\chi,k)|^{2}+O(q^{4/10^{M}}).\end{split}

Denote $\phi(q)$ the Euler totient function, we recall that the orthogonality relation for Dirichlet characters (see [MVa1, Corollary 4.5]) asserts

(2.10)

\displaystyle\sum_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\chi(n)=\begin{cases}\varphi(q)\quad\text{if}\ n\equiv 1\pmod{q},\\ 0\quad\text{otherwise}.\end{cases}

As the length of the Dirichlet series ${\mathcal{N}}(\chi,k)$ is $\leq q$ , we deduce from (2.10) that only the diagonal terms in the last sum of (2.9) survive. Thus, we have

\displaystyle\begin{split}&\sum_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|{\mathcal{N}}(\chi,k)|^{2}\geq\phi(q)\prod^{R}_{j=1}\Big{(}\sum_{n_{j}}\frac{k^{2\Omega(n_{j})}}{n_{j}w^{2}(n_{j})}b_{j}(n_{j})\Big{)}\geq\phi^{*}(q)\prod^{R}_{j=1}\Big{(}\sum_{n_{j}}\frac{k^{2\Omega(n_{j})}}{n_{j}w^{2}(n_{j})}b_{j}(n_{j})\Big{)}\gg\phi^{*}(q)(\log q)^{k^{2}},\end{split}

where the last estimation above follows from [Gao2024-6, (6.3)] and the treatments given in [Gao2024-6, Section 6]. This together with (2.9) now implies the assertions of Proposition 2.2.

2.6. Proof of Proposition 2.3

We notice that it follows from [Gao2024-6, (3.5)] that we have for $|z|\leq aK/10$ with $0<a\leq 1$ ,

(2.11)

\displaystyle\Big{|}\sum_{r=0}^{K}\frac{z^{r}}{r!}-e^{z}\Big{|}\leq\frac{|z|^{K}}{K!}\leq\Big{(}\frac{ae}{10}\Big{)}^{K}.

For any fixed $1\leq j\leq R$ , we apply (2.11) with $z=k{\mathcal{P}}_{j}(\chi),K=\ell_{j}$ and $a=1$ to see that when $|{\mathcal{P}}_{j}(\chi)|\leq\ell_{j}/(10(1+|k|))$ ,

(2.12)

\displaystyle{\mathcal{N}}_{j}(\chi,k)=

\displaystyle\exp(k{\mathcal{P}}_{j}(\chi))\Big{(}1+O\Big{(}\exp(|k{\mathcal{P}}_{j}(\chi)|)\Big{(}\frac{e}{10}\Big{)}^{\ell_{j}}\Big{)}=\exp(k{\mathcal{P}}_{j}(\chi))\Big{(}1+O\Big{(}e^{-\ell_{j}}\Big{)}\Big{)}.

Similarly, when $|{\mathcal{P}}_{j}(\chi)|\leq\ell_{j}/10(1+|k|)$ , we have

(2.13)

\displaystyle{\mathcal{M}}_{j}(\chi,1-k)=

\displaystyle\exp((1-k)\Re{\mathcal{P}}_{j}(\chi))\Big{(}1+O\Big{(}e^{-\ell_{j}}\Big{)}\Big{)}.

It follows from (2.12) and (2.13) that when $|{\mathcal{P}}_{j}(\chi)|\leq\ell_{j}/(10(1+|k|))$ ,

(2.14)

\displaystyle\Big{|}\mathcal{N}_{j}(\chi,k)^{2}\mathcal{M}_{j}(\chi,1-k)^{-k/(1-k)}\Big{|}^{2}=

\displaystyle\exp(2k\Re({\mathcal{P}}_{j}(\chi)))\Big{(}1+O\big{(}e^{-\ell_{j}}\big{)}\Big{)}=|{\mathcal{N}}_{j}(\chi,k)|^{2}\Big{(}1+O\big{(}e^{-\ell_{j}}\big{)}\Big{)}.

On the other hand, we notice that when $|{\mathcal{P}}_{j}(\chi)|\geq\ell_{j}/(10(1+|k|))$ ,

(2.15)

\displaystyle\begin{split}|{\mathcal{N}}_{j}(\chi,k)|&\leq\sum_{r=0}^{\ell_{j}}\frac{|k{\mathcal{P}}_{j}(\chi)|^{r}}{r!}\leq|{\mathcal{P}}_{j}(\chi)|^{\ell_{j}}\sum_{r=0}^{\ell_{j}}\Big{(}\frac{10(1+|k|)}{\ell_{j}}\Big{)}^{\ell_{j}-r}\frac{|k|^{r}}{r!}\leq\Big{(}\frac{12(1+|k|)|{\mathcal{P}}_{j}(\chi)|}{\ell_{j}}\Big{)}^{\ell_{j}}.\end{split}

Notice that the same bound above holds for $|{\mathcal{M}}_{j}(\chi,1-k)|$ as well. We then deduce from these estimations that when $|{\mathcal{P}}_{j}(\chi)|\geq\ell_{j}/(10(1+|k|))$ , we have

(2.16)

\displaystyle\Big{|}\mathcal{N}_{j}(\chi,k)^{2}\mathcal{M}_{j}(\chi,1-k)^{-k/(1-k)}\Big{|}^{2}

\displaystyle\leq\Big{(}\frac{12(1+|k|)|{\mathcal{P}}_{j}(\chi)|}{\ell_{j}}\Big{)}^{2(2-k/(1-k))\ell_{j}}\leq|{\mathcal{Q}}_{j}(\chi,k)|^{2}.

It follows (2.2), (2.3), (2.14) and (2.16) that we have

	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{\|}\mathcal{N}(\chi,k)^{2}\mathcal{M}(\chi,1-k)^{-k/(1-k)}\Big{\|}^{2}\ll$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\prod^{R}_{j=1}\Big{(}\|{\mathcal{N}}_{j}(\chi,k)\|^{2}\Big{(}1+O\big{(}e^{-\ell_{j}}\big{)}\Big{)}+\|{\mathcal{Q}}_{j}(\chi,k)\|^{2}\Big{)}$
	$\displaystyle\ll$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\prod^{R}_{j=1}\Big{(}1+O(e^{-\ell_{j}/2})\Big{)}\prod^{R}_{j=1}\Big{(}\|{\mathcal{N}}_{j}(\chi,k)\|^{2}+\|{\mathcal{Q}}_{j}(\chi,k)\|^{2}\Big{)}$
	$\displaystyle\ll$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\prod^{R}_{j=1}\Big{(}\|{\mathcal{N}}_{j}(\chi,k)\|^{2}+\|{\mathcal{Q}}_{j}(\chi,k)\|^{2}\Big{)},$

where the last estimation above follows from the observation that the sum over $e^{-\ell_{j}/2}$ converges. Now, a straightforward adaption of the proof of [Gao2024-6, Proposition 3.5] implies that the last expression above is $\ll\phi^{*}(q)(\log q)^{k^{2}}$ . This completes the proof of Proposition 2.3.

2.7. Proof of Proposition 2.4

Note that similar estimations for ${\mathcal{N}}_{j}(\chi,k)$ given in (2.12) and (2.15) are valid for $\mathcal{M}_{j}(\chi,k-1)$ as well. We thus conclude that for each $1\leq j\leq R$ ,

\displaystyle\begin{split}|{\mathcal{M}}_{j}(\chi,k-1)|^{2}\leq&|{\mathcal{N}}_{j}(\chi,k-1)|^{2}\Big{(}1+O\Big{(}e^{-\ell_{j}}\Big{)}\Big{)}+|{\mathcal{Q}}_{j}(\chi,k)|^{2}\\ \leq&\Big{(}1+O\Big{(}e^{-\ell_{j}}\Big{)}\Big{)}\Big{(}|{\mathcal{N}}_{j}(\chi,k-1)|^{2}+|{\mathcal{Q}}_{j}(\chi,k)|^{2}\Big{)}.\end{split}

It follows that

(2.17)

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\big{|}L(\tfrac{1}{2},\chi){\mathcal{M}}(\chi,k-1)\big{|}^{2}\leq&\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\prod^{R}_{j=1}\Big{(}1+O(e^{-\ell_{j}/2})\Big{)}\big{|}L(\tfrac{1}{2},\chi)\big{|}^{2}\prod^{R}_{j=1}\Big{(}|{\mathcal{N}}_{j}(\chi,k-1)|^{2}+|{\mathcal{Q}}_{j}(\chi,k)|^{2}\Big{)}\\ \leq&\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\big{|}L(\tfrac{1}{2},\chi)\big{|}^{2}\prod^{R}_{j=1}\Big{(}|{\mathcal{N}}_{j}(\chi,k-1)|^{2}+|{\mathcal{Q}}_{j}(\chi,k)|^{2}\Big{)}\\ \leq&\sum_{S,S^{c}}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\big{|}L(\tfrac{1}{2},\chi)\big{|}^{2}\prod_{j\in S,i\in S^{c}}|{\mathcal{N}}_{j}(\chi,k-1)|^{2}|{\mathcal{Q}}_{i}(\chi,k)|^{2},\end{split}

where the sum $\sum_{S}$ is over all subsets $S$ of the set $\{1,\cdots,R\}$ and where we denote $S^{c}$ for the complement of $S$ in $\{1,\cdots,R\}$ . Now, an inspection of the proof of [Gao2024-6, Proposition 3.4] implies that for any fixed such $S$ , we have

\displaystyle\big{|}L(\tfrac{1}{2},\chi)\big{|}^{2}\prod_{j\in S,i\in S^{c}}|{\mathcal{N}}_{j}(\chi,k-1)|^{2}|{\mathcal{Q}}_{i}(\chi,k)|^{2}\ll\exp(-\sum_{i\in S^{c}}\frac{\ell_{i}}{2})\phi^{*}(q)(\log q)^{k^{2}}.

We deduce from this and (2.17) that

\displaystyle\begin{split}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\big{|}L(\tfrac{1}{2},\chi){\mathcal{M}}(\chi,k-1)\big{|}^{2}\ll&\prod^{R}_{j=1}\Big{(}1+O(e^{-\ell_{j}/2})\Big{)}\phi^{*}(q)(\log q)^{k^{2}}\ll\phi^{*}(q)(\log q)^{k^{2}}.\end{split}

This establishes the desired estimation given in Proposition 2.4 and hence completes the proof.

Acknowledgments. The author is supported in part by NSFC grant 11871082.

	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\Big{\|}\mathcal{N}(\chi,k)^{2}\mathcal{M}(\chi,1-k)^{-k/(1-k)}\Big{\|}^{2}\ll$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\prod^{R}_{j=1}\Big{(}\|{\mathcal{N}}_{j}(\chi,k)\|^{2}\Big{(}1+O\big{(}e^{-\ell_{j}}\big{)}\Big{)}+\|{\mathcal{Q}}_{j}(\chi,k)\|^{2}\Big{)}$
	$\displaystyle\ll$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\prod^{R}_{j=1}\Big{(}1+O(e^{-\ell_{j}/2})\Big{)}\prod^{R}_{j=1}\Big{(}\|{\mathcal{N}}_{j}(\chi,k)\|^{2}+\|{\mathcal{Q}}_{j}(\chi,k)\|^{2}\Big{)}$
	$\displaystyle\ll$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\prod^{R}_{j=1}\Big{(}\|{\mathcal{N}}_{j}(\chi,k)\|^{2}+\|{\mathcal{Q}}_{j}(\chi,k)\|^{2}\Big{)},$

Lower bounds for negative moments of Dirichlet LL-functions to a fixed modulus

Abstract.

1. Introduction

Theorem 1.1.

2. Proof of Theorem 1.1

2.1. Setup

Proposition 2.2.

Proposition 2.3.

Proposition 2.4.

2.5. Proof of Proposition 2.2

2.6. Proof of Proposition 2.3

2.7. Proof of Proposition 2.4

References

Lower bounds for negative moments of Dirichlet $L$ -functions to a fixed modulus