Bounds for moments of Dirichlet $L$-functions to a fixed modulus

A considerable amount of work in the literature has been done on moments of central values of families of $L$-functions, due to rich arithmetic meanings these central values have. In this paper, we focus on the family of Dirichlet $L$-functions to a fixed modulus. It is widely believed that (see [R&Sound]) for all real $k\geq 0$ and large integers $q\not\equiv 2\pmod{4}$ (so that primitive Dirichlet characters modulo $q$ exist),

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

The formula given in (1.1) is well-known for $k=1$ and is a conjecture due to K. Ramachandra [Rama79] for $k=2$ when the sum in (1.1) is being replaced by the sum over all Dirichlet characters modulo a prime $q$. For all most all $q$, D. R. Heath-Brown [HB81] established (1.1) for $k=2$ and K. Soundararajan [Sound2007] improved the result to be valid for all $q$. An asymptotic formula with a power saving error term was further obtained in this case for $q$ being prime numbers by M. P. Young [Young2011]. The main terms in Young’s result agree with a conjectured formula provided by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein and N. C. Snaith in [CFKRS] concerning the left side of (1.1) for all positive integral values of $k$. Subsequent improvements on the error terms in Young’s result are given in [BFKMM1] and [BFKMM]. See also [Wu2020] for an extension of Young’s result to general moduli.

Other than the asymptotic relations given in (1.1), much is known on upper and lower bounds of the conjectured order of magnitude for moments of the family of $L$-functions under consideration. To give an account for the related results, we assume that $q$ is a prime number in the rest of the paper. In [Sound01], under the assumption of the generalized Riemann hypothesis (GRH), K. Soundararajan showed that

$$\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}|L(\tfrac{1}{2},\chi)|^{2k}\ll_{k}\phi^{*}(q)(\log q)^{k^{2}+\varepsilon}$$

for all real positive $k$ and any $\varepsilon>0$. These bounds are optimal except for the $\varepsilon$ powers. The optimal upper bounds are later obtained by D. R. Heath-Brown in [HB2010] unconditionally for $k=1/v$ with $v$ a positive integer and for all $k\in(0,2)$ under GRH. Using a sharpening of the method of Soundararajan by A. J. Harper in [Harper], one may also establish the optimal upper bounds for all real $k\geq 0$ under GRH. In [Radziwill&Sound], M. Radziwiłł and K. Soundararajan enunciated a principle that allows one to establish sharp upper bounds for moments of families of $L$-functions unconditionally and used it to study the moments of quadratic twists of $L$-functions attached to elliptic curves. This principle was then applied by W. Heap, M. Radziwiłł and K. Soundararajan in [HRS] to establish unconditionally the $2k$-th moment of the Riemann zeta function on the critical line for all real $0\leq k\leq 2$.

In the opposite direction, a simple and powerful method developed by Z. Rudnick and K. Soundararajan in [R&Sound1] shows that

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

for all rational $k\geq 1$. A modification of a method of M. Radziwiłł and K. Soundararajan in [Radziwill&Sound1] may allow one to establish such lower bounds for all real $k\geq 1$. In [C&L], V. Chandee and X. Li obtained the above lower bounds for rational $0<k<1$.

In [H&Sound], W. Heap and K. Soundararajan developed another principle which allows one to study lower bounds of families of $L$-functions. This principle can be regarded as a companion to the above principle of M. Radziwiłł and K. Soundararajan [Radziwill&Sound] concerning upper bounds. Although Heap and Soundararajan only studied moments of the Riemann zeta function on the critical line, they did point out that their principle may be applied to study moments of families of $L$-functions, including the one we consider in this paper. In fact, the density conjecture of N. Katz and P. Sarnak concerning low-lying zeros of families of $L$-functions indicates that the underlying symmetry for the family of Dirichlet $L$-functions to a fixed modulus is unitary, and that the behaviour of this family resembles that of the Riemann zeta function on the critical line. Thus, one expects to obtain sharp lower bounds for moments of the above unitary family of $L$-functions using the principle of Heap and Soundararajan. The aim of this paper is to first carry out this principle explicitly to achieve the desired lower bounds in the following result.

Next, we apply the dual principle of M. Radziwiłł and K. Soundararajan [Radziwill&Sound] to establish sharp upper bounds for a restricted range of $k$ as follows.

Note that we can combine Theorem 1.1 and 1.2 together to obtain the following result concerning the order of magnitude of our family of $L$-functions.

We notice that such a result in (1.4) is already implied by the above mentioned result of D. R. Heath-Brown [HB2010] and that of V. Chandee and X. Li [C&L]. In particular, the case $k=1$ of (1.4) is explicitly given in [C&L]. Moreover, the result is shown to be valid for $k=3/2$ as well by H. M. Bui, K. Pratt, N. Robles and A. Zaharescu [BPRZ, Theorem 1.4]. This case is achieved by employing various tools including a result on a long mollified second moment of the corresponding family of $L$-functions given in [BPRZ, Theorem 1.1]. In our proofs of Theorems 1.1 and 1.2, we also need to evaluate certain twisted second moments for the same family. The lengths of the corresponding Dirichlet polynomials are however short so that the main contributions come only from the diagonal terms. Hence, only the orthogonality relation for characters is needed to complete our work. We also point out here that as it is mentioned in [BPRZ] that one may apply the work of B. Hough [Hough2016] or R. Zacharias [Z2019] on twisted fourth moment for the family of Dirichlet $L$-functions modulo $q$ to obtain sharp upper bounds on all moments below the fourth. We decide to use the twisted second moment here to keep our exposition simple by observing that it is needed for obtaining both the lower bounds and the upper bounds.

We include a few auxiliary results in this section. We also reserve the letter $p$ for a prime number in this paper and we recall the following result from [Gao2021-2, Lemma 2.2].

Next, we note the following approximate functional equation for $|L(1/2,\chi)|^{2}$.

The above lemma follows by combining equations [Sound2007, (1.2)-(1.4)] and Lemma 2 there, together with the observation that the property that $\mathcal{W}_{\mathfrak{a}}(x)$ is real valued can be established similar to [sound1, Lemma 2.1].

The presence of $\mathcal{W}_{\mathfrak{a}}(x)$ in the expression for $|L(1/2,\chi)|^{2}$ makes it natural to consider sums over odd and even characters separately when summing over $\chi$ modulo $q$. For this reason, we denote $\phi(q)$ for the Euler totient function and note the following orthogonal relations.

We may assume that $q$ is a large prime number and we note that in this case we have $\phi^{*}(q)=q-2$. As the case $k=1$ for both (1.2) and (1.3) is known, we may assume in our proofs that $k\neq 1$ is a fixed positive real number and let $N,M$ be two large natural numbers depending on $k$ only and and denote $\{\ell_{j}\}_{1\leq j\leq R}$ for a sequence of even natural numbers 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 $R$ is defined to the largest natural number satisfying $\ell_{R}>10^{M}$. We may assume that $M$ is so chosen so that we have $\ell_{j}>\ell_{j+1}^{2}$ for all $1\leq j\leq R-1$ and this further implies that we have

(3.1)

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

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

$${\mathcal{P}}_{j}(\chi)=\sum_{p\in P_{j}}\frac{1}{\sqrt{p}}\chi(p),\quad{\mathcal{Q}}_{j}(\chi,k)=\Big{(}\frac{12\max(1,k^{2}){\mathcal{P}}(\chi)}{\ell_{j}}\Big{)}^{r_{k}\ell_{j}},$$

where we define $r_{k}=\lceil 1+1/k\rceil+1$ for $0<k<1$ and $r_{k}=\lceil k/(2k-1)\rceil+1$ for $k>1$. We further define ${\mathcal{Q}}_{R+1}(\chi,k)=1$.

We define for any non-negative integer $\ell$ and any real 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 $\alpha$,

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

Before we proceed to our discussions below, we would like to point out here without further notice that in the rest of the paper, when we use $\ll$ or the $O$-symbol to estimate various quantities needed, the implicit constants involved only depend on $k$ and are uniform with respect to $\chi$. We shall also make the convention that an empty product is defined to be $1$.

We now present the needed versions in our setting of the lower bounds principle of W. Heap and K. Soundararajanand in [H&Sound] and the upper bounds principle of M. Radziwiłł and K. Soundararajan in [Radziwill&Sound] in the following two lemmas. We choose to state our results suitable for our proofs of Theorems 1.1 and 1.2 only. One may easily adjust them to study moments for various other families of $L$-functions.

Our first lemma corresponds to the lower bounds principle.

Our next lemma corresponds to the upper bounds principle. Instead of the form used for obtaining upper bounds given in [Radziwill&Sound], we decide to adapt one that resembles what is given in Lemma 3.1 above and also derive it via a similar fashion. One may compare our next lemma with [Radziwill&Sound, Proposition 3] and [HRS, Proposition 2.1].

In what follows, we may further assume that $0<k<1$ by noting that the case $k\geq 1$ of Theorem 1.1 can be obtained using (3.3) in Lemma 3.1 and applying the arguments in the paper. We then deduce from Lemma 3.1 and Lemma 3.2 that in order to prove Theorem 1.1 and 1.2 for the case $0<k<1$, it suffices to establish the following three propositions.

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

Denote $\Omega(n)$ for the number of distinct prime powers dividing $n$ and $w(n)$ for the multiplicative function such that $w(p^{\alpha})=\alpha!$ for prime powers $p^{\alpha}$. Let $b_{j}(n),1\leq j\leq R$ be functions such that $b_{j}(n)=1$ when $n$ is composed of at most $\ell_{j}$ primes, all from the interval $P_{j}$. Otherwise, we define $b_{j}(n)=0$. We use these notations to see that for any real number $\alpha$,

(4.1)

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

Note that each ${\mathcal{N}}_{j}(\chi,\alpha)$ is a short Dirichlet polynomial since $b_{j}(n_{j})=0$ unless $n_{j}\leq(q^{1/\ell_{j}^{2}})^{\ell_{j}}=q^{1/\ell_{j}}$. It follows from this that ${\mathcal{N}}(\chi,k)$ and ${\mathcal{N}}(\chi,k-1)$ are short Dirichlet polynomials whose lengths are both at most $q^{1/\ell_{1}+\ldots+1/\ell_{R}}<q^{2/10^{M}}$ by (3.1). Moreover, it is readily checked that we have for each $\chi$ modulo $q$ (including the case $\chi=\chi_{0}$, the principal character modulo $q$),

(4.2)

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

We note further that it is shown in [R&Sound] that for $X\geq 1$,

$\displaystyle L(\tfrac{1}{2},\chi)=\sum_{m\leq X}\frac{\chi(m)}{\sqrt{m}}+O(\frac{\sqrt{q}\log q}{\sqrt{X}}).$

We deduce from the above that

		$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}L(\tfrac{1}{2},\chi)\mathcal{N}(\overline{\chi},k)\mathcal{N}(\chi,k-1)$
	$\displaystyle=$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\sum_{m\leq X}\frac{\chi(m)}{\sqrt{m}}\mathcal{N}(\overline{\chi},k)\mathcal{N}(\chi,k-1)+O(\frac{\sqrt{q}\log q}{\sqrt{X}}\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\mathcal{N}(\overline{\chi},k)\mathcal{N}(\chi,k-1))$
	$\displaystyle=$	$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\sum_{m\leq X}\frac{\chi(m)}{\sqrt{m}}\mathcal{N}(\overline{\chi},k)\mathcal{N}(\chi,k-1)+O(\frac{\phi^{*}(q)q^{1/2+4/10^{M}}\log q}{\sqrt{X}}),$

where the last estimation above follows from (4.2). Applying (4.2) one more time, we see that the main term above equals

		$\displaystyle\sum_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}\sum_{m\leq X}\frac{\chi(m)}{\sqrt{m}}\mathcal{N}(\overline{\chi},k)\mathcal{N}(\chi,k-1)+O(\sqrt{X}q^{4/10^{M}})$
	$\displaystyle=$	$\displaystyle\phi^{*}(q)\sum_{a}\sum_{b}\sum_{\begin{subarray}{c}n\leq X\\ an\equiv b\bmod q\end{subarray}}\frac{x_{a}y_{b}}{\sqrt{abn}}+O(\sqrt{X}q^{4/10^{M}}),$

where we write for simplicity

$\displaystyle{\mathcal{N}}(\chi,k-1)=\sum_{a\leq q^{2/10^{M}}}\frac{x_{a}}{\sqrt{a}}\chi(a),\quad\mathcal{N}(\overline{\chi},k)=\sum_{b\leq q^{2/10^{M}}}\frac{y_{b}}{\sqrt{b}}\overline{\chi}(b).$

We now consider the contribution from the terms $am=b+lq$ with $l\geq 1$ above (note that as $b<q$, we can not have $b>am$ in our case). As this implies that $l\leq q^{2/10^{M}}X/q$, we deduce together with the observation that $x_{a},y_{b}\ll 1$ that the total contribution from these terms is

$\displaystyle\ll$

$\displaystyle\phi(q)\sum_{b\leq q^{2/10^{M}}}\sum_{l\leq q^{2/10^{M}}X/q}\frac{1}{\sqrt{bql}}\ll\sqrt{X}q^{2/10^{M}}.$

We now set $X=q^{1+1/10^{M-1}}$ to see that we can ignore the contributions from various error terms above to deduce that

$\displaystyle\sideset{}{{}^{*}}{\sum}_{\begin{subarray}{c}\chi\negthickspace\negthickspace\negthickspace\pmod{q}\end{subarray}}L(\tfrac{1}{2},\chi)\mathcal{N}(\overline{\chi},k)\mathcal{N}(\chi,k-1)\gg\phi^{*}(q)\sum_{a}\sum_{b}\sum_{\begin{subarray}{c}m\leq X\\ am=b\end{subarray}}\frac{x_{a}y_{b}}{\sqrt{abm}}=\phi^{*}(q)\sum_{b}\frac{y_{b}}{b}\sum_{\begin{subarray}{c}a,m\\ am=b\end{subarray}}x_{a}=\phi^{*}(q)\sum_{b}\frac{y_{b}}{b}\sum_{\begin{subarray}{c}a|b\end{subarray}}x_{a},$

where the last equality above follows from the observation that $b\leq q^{2/10^{M}}<X$.

Notice that

(4.3)

$\displaystyle\begin{split}\sum_{b}\frac{y_{b}}{b}\sum_{\begin{subarray}{c}a|b\end{subarray}}x_{a}=&\prod^{R}_{j=1}\Big{(}\sum_{n_{j}}\frac{1}{n_{j}}\frac{k^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\sum_{n^{\prime}_{j}|n_{j}}\frac{(k-1)^{\Omega(n^{\prime}_{j})}}{w(n^{\prime}_{j})}b_{j}(n^{\prime}_{j})\Big{)}\\ =&\prod^{R}_{j=1}\Big{(}\sum_{n_{j}}\frac{1}{n_{j}}\frac{k^{\Omega(n_{j})}}{w(n_{j})}b_{j}(n_{j})\sum_{n^{\prime}_{j}|n_{j}}\frac{(k-1)^{\Omega(n^{\prime}_{j})}}{w(n^{\prime}_{j})}\Big{)},\end{split}$

where the last equality above follows by noting that $b_{j}(n_{j})=1$ implies that $b_{j}(n^{\prime}_{j})=1$ for all $n^{\prime}_{j}|n_{j}$.