1 Introduction
Dirichlet characters of a fixed order appear naturally in many applications in number theory. The quadratic characters have been extensively studied, due in large part to their connection to fundamental arithmetic objects including class numbers and ranks of elliptic curves. By contrast, higher-order characters have received considerably less attention up until very recently, when Granville and Soundararajan [Reference Granville and Soundararajan14] proved the remarkable result that the classical Pólya–Vinogradov inequality can be improved for characters of a fixed odd order. Other notable work on higher-order characters includes the results of Baier and Young [Reference Baier and Young1] on large sieve inequalities and moments of Dirichlet
$L$
-functions attached to cubic characters, which build on previous work by Heath-Brown [Reference Heath-Brown15] on cubic symbols; the large sieve inequalities for quartic characters by Gao and Zhao [Reference Gao and Zhao8]; and the work of Blomer et al [Reference Blomer, Goldmakher and Louvel3] on large sieve inequalities and double Dirichlet series associated with certain higher-order Hecke characters.
The connection between Dirichlet characters and class numbers was discovered by Dirichlet in 1839, who established a formula that expresses the class number of a quadratic field
$\mathbb{Q}(\sqrt{d})$
in terms of
$L(1,\unicode[STIX]{x1D712}_{d})$
, the value of the corresponding Dirichlet
$L$
-function at one, where
$\unicode[STIX]{x1D712}_{d}:=(\frac{d}{\cdot })$
is the Kronecker symbol. Motivated by Dirichlet’s class number formula, Littlewood [Reference Littlewood16] studied how large can
$L(1,\unicode[STIX]{x1D712})$
be in terms of the conductor of
$\unicode[STIX]{x1D712}$
. Assuming the generalized Riemann hypothesis (GRH), Littlewood proved that, for any non-principal primitive character
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
,
$$\begin{eqnarray}|L(1,\unicode[STIX]{x1D712})|\leqslant (2\text{e}^{\unicode[STIX]{x1D6FE}}+o(1))\log \log q.\end{eqnarray}$$
On the other hand, under the same hypothesis, Littlewood [Reference Littlewood16] showed that there exist infinitely many fundamental discriminants
$d$
(both positive and negative) for which
$$\begin{eqnarray}L(1,\unicode[STIX]{x1D712}_{d})\geqslant (\text{e}^{\unicode[STIX]{x1D6FE}}+o(1))\log \log |d|.\end{eqnarray}$$
This omega result was later established unconditionally by Chowla [Reference Chowla6].
To understand which of the bounds (1.1) or (1.2) is closer to the maximal values of
$L(1,\unicode[STIX]{x1D712}_{d})$
, Montgomery and Vaughan [Reference Montgomery and Vaughan18] constructed a probabilistic random model for these values and made several conjectures on their distribution. Most of these conjectures were subsequently proved by Granville and Soundararajan [Reference Granville and Soundararajan12]. Among their results, Granville and Soundararajan obtained an asymptotic formula for the distribution function of
$L(1,\unicode[STIX]{x1D712}_{d})$
, showing that the tail of this distribution is double exponentially decreasing. In particular, their work gives strong support to the conjecture that Chowla’s omega result (1.2) corresponds to the true nature of extreme values of
$L(1,\unicode[STIX]{x1D712}_{d})$
. In [Reference Granville and Soundararajan13], Granville and Soundararajan proved similar results for the distribution of the values
$|L(1,\unicode[STIX]{x1D712})|$
as
$\unicode[STIX]{x1D712}$
varies over the non-principal primitive characters modulo
$q$
with
$q\leqslant Q$
. Their results give very solid evidence for the following widely believed conjecture.
Conjecture 1.1. Let
$Q$
be large. Then
$$\begin{eqnarray}\max _{q\leqslant Q}~\max _{\substack{ \unicode[STIX]{x1D712}\neq \unicode[STIX]{x1D712}_{0}~(\text{mod}~q) \\ \unicode[STIX]{x1D712}~\text{primitive}}}|L(1,\unicode[STIX]{x1D712})|=(\text{e}^{\unicode[STIX]{x1D6FE}}+o(1))\log \log Q.\end{eqnarray}$$
Despite all the progress made on
$L(1,\unicode[STIX]{x1D712}_{d})$
, very little is known on the values of
$L(1,\unicode[STIX]{x1D712})$
for higher-order characters
$\unicode[STIX]{x1D712}$
. The main difficulty is that, unlike the quadratic case where one is equipped with the powerful law of quadratic reciprocity, higher reciprocity laws for
$k$
th order symbols are not easy to apply to Dirichlet characters of order
$k$
.
In this paper, we exhibit large values of
$|L(1,\unicode[STIX]{x1D712})|$
for
$k$
th order characters
$\unicode[STIX]{x1D712}$
. We also apply our work to improve recent results of Goldmakher and Lamzouri [Reference Goldmakher and Lamzouri11] and Bober [Reference Bober4], by obtaining lower bounds for even-order character sums, which are likely to be optimal. Our first theorem extends Chowla’s omega result (1.2) to characters of any given order
$k$
. In view of Conjecture 1.1, we believe our bound to be the best possible.
Theorem 1.2. Let
$k\geqslant 2$
be fixed and let
$Q$
be large. There exists a constant
$c_{k}>0$
for which there are at least
$Q\exp (-c_{k}\log Q/\text{log}\log Q)$
primitive characters
$\unicode[STIX]{x1D712}$
of order
$k$
and conductor
$q\leqslant Q$
such that
$$\begin{eqnarray}|L(1,\unicode[STIX]{x1D712})|\geqslant \text{e}^{\unicode[STIX]{x1D6FE}}\log \log Q+O_{k}(1).\end{eqnarray}$$
Let
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
be a non-principal primitive character. An important quantity attached to
$\unicode[STIX]{x1D712}$
is
$$\begin{eqnarray}M(\unicode[STIX]{x1D712}):=\max _{x}\biggl|\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D712}(n)\biggr|.\end{eqnarray}$$
This character sum has been extensively studied over the past century. The first non-trivial bound on
$M(\unicode[STIX]{x1D712})$
, proved independently by Pólya [Reference Pólya21] and Vinogradov [Reference Vinogradov22] in 1918, asserts that
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\ll \sqrt{q}\log q.\end{eqnarray}$$
This upper bound remains the strongest known outside of special cases. However, conditionally on the GRH, Montgomery and Vaughan [Reference Montgomery and Vaughan17] proved that
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\ll \sqrt{q}\log \log q.\end{eqnarray}$$
Recently, in a groundbreaking paper [Reference Granville and Soundararajan14], Granville and Soundararajan improved both the Pólya–Vinogradov inequality and the Montgomery–Vaughan GRH bound for characters of a given odd order. More precisely, they showed that if
$g\geqslant 3$
is an odd integer and
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
is a primitive character of order
$g$
, then
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\ll \left\{\begin{array}{@{}ll@{}}\displaystyle \sqrt{q}(\log q)^{1-\unicode[STIX]{x1D6FF}_{g}/2+o(1)}\quad & \text{ unconditionally},\\ \displaystyle \sqrt{q}(\log \log q)^{1-\unicode[STIX]{x1D6FF}_{g}/2+o(1)}\quad & \text{ under GRH},\end{array}\right.\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FF}_{g}:=1-(g/\unicode[STIX]{x1D70B})\sin (\unicode[STIX]{x1D70B}/g)$
. (Here
$o(1)\rightarrow 0$
as
$q\rightarrow \infty$
.) After developing their ideas further, Goldmakher [Reference Goldmakher9] showed that these bounds hold with the exponent
$1-\unicode[STIX]{x1D6FF}_{g}$
instead of
$1-\unicode[STIX]{x1D6FF}_{g}/2$
, and hence
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\ll \sqrt{q}(\log \log q)^{(g/\unicode[STIX]{x1D70B})\sin (\unicode[STIX]{x1D70B}/g)+o(1)},\end{eqnarray}$$
under the assumption of GRH. Shortly afterwards, Goldmakher and Lamzouri [Reference Goldmakher and Lamzouri10] proved that this bound is, in fact, optimal. More precisely, they showed that for any
$\unicode[STIX]{x1D716}>0$
and any fixed odd integer
$g\geqslant 3$
, there exist arbitrarily large
$q$
and primitive characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
of order
$g$
satisfying
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\gg _{g,\unicode[STIX]{x1D716}}\sqrt{q}(\log \log q)^{(g/\unicode[STIX]{x1D70B})\sin (\unicode[STIX]{x1D70B}/g)-\unicode[STIX]{x1D716}}.\end{eqnarray}$$
The situation for even-order characters is completely different. Indeed, an old result of Paley [Reference Paley20] asserts the existence of an infinite family of quadratic characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
for which
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\gg \sqrt{q}\log \log q,\end{eqnarray}$$
showing that the Montgomery–Vaughan bound (1.3) is sharp in this case. This was extended to characters of a given even-order by Granville and Soundararajan [Reference Granville and Soundararajan14] under the assumption of GRH. Recently, Goldmakher and Lamzouri [Reference Goldmakher and Lamzouri11] obtained this result unconditionally.
Granville and Soundararajan [Reference Granville and Soundararajan14] also refined the GRH bound of Montgomery and Vaughan (1.3) for all characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
. More specifically, assuming the GRH, they showed that
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\leqslant \left\{\begin{array}{@{}ll@{}}\displaystyle \biggl(\frac{2\text{e}^{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D70B}}+o(1)\biggr)\sqrt{q}\log \log q\quad & \text{ if }\unicode[STIX]{x1D712}\text{ is odd }(\text{that is }\unicode[STIX]{x1D712}(-1)=-1),\\ \displaystyle \biggl(\frac{2\text{e}^{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D70B}\sqrt{3}}+o(1)\biggr)\sqrt{q}\log \log q\quad & \text{ if }\unicode[STIX]{x1D712}\text{ is even }(\text{that is }\unicode[STIX]{x1D712}(-1)=1).\end{array}\right.\end{eqnarray}$$
Similarly to the case of
$L(1,\unicode[STIX]{x1D712})$
, Granville and Soundararajan [Reference Granville and Soundararajan14] conjectured that the GRH bounds (1.6) are off by a factor of two.
Conjecture 1.3. Let
$q$
be large, and
$\unicode[STIX]{x1D712}$
be a primitive character
$(\text{mod}\;q)$
. Then
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\leqslant \left\{\begin{array}{@{}ll@{}}\displaystyle \biggl(\frac{\text{e}^{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D70B}}+o(1)\biggr)\sqrt{q}\log \log q\quad & \text{ if }\unicode[STIX]{x1D712}\text{ is odd, }\\ \displaystyle \biggl(\frac{\text{e}^{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D70B}\sqrt{3}}+o(1)\biggr)\sqrt{q}\log \log q\quad & \text{ if }\unicode[STIX]{x1D712}\text{ is even}.\end{array}\right.\end{eqnarray}$$
In a recent work, Bober et al [Reference Bober, Goldmakher, Granville and Koukoulopoulos5] studied the distribution of large values of
$M(\unicode[STIX]{x1D712})$
as
$\unicode[STIX]{x1D712}$
varies over non-principal characters modulo
$q$
, where
$q$
is a large prime. Among their results, they showed that the bounds in (1.7) are the best possible. Indeed, they proved that if
$q$
is a large prime, then there exist non-principal even and odd characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
such that
$M(\unicode[STIX]{x1D712})\geqslant (c+o(1))\sqrt{q}\log \log q$
, where
$c=\text{e}^{\unicode[STIX]{x1D6FE}}/\unicode[STIX]{x1D70B}$
if
$\unicode[STIX]{x1D712}$
is odd and
$c=\text{e}^{\unicode[STIX]{x1D6FE}}/(\unicode[STIX]{x1D70B}\sqrt{3})$
if
$\unicode[STIX]{x1D712}$
is even. Moreover, their Theorems 1.1 and 1.3 give strong support to Conjecture 1.3.
In the case of quadratic characters, Bateman and Chowla [Reference Bateman and Chowla2] improved Paley’s result (1.5) by establishing the existence of infinitely many
$q$
and odd quadratic characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
such that
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\geqslant \biggl(\frac{\text{e}^{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D70B}}+o(1)\biggr)\sqrt{q}\log \log q.\end{eqnarray}$$
In view of Conjecture 1.3, this bound is likely to be the best possible. When
$k\geqslant 4$
is an even integer, Goldmakher and Lamzouri [Reference Goldmakher and Lamzouri11] extended Paley’s construction to characters of order
$k$
. More precisely, they showed that there are infinitely many
$q$
and even characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
of order
$k$
such that
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\geqslant \biggl(\frac{1}{\unicode[STIX]{x1D70B}\sqrt{p_{k}}}+o(1)\biggr)\sqrt{q}\log \log q,\end{eqnarray}$$
where
$p_{k}$
is the smallest prime such that
$p_{k}\equiv k+1~(\text{mod}\;2k)$
. Since
$p_{k}>k$
, this bound decreases as the order increases. Bober [Reference Bober4] subsequently obtained the same result using a different approach. As Bober notes that, this leaves open the possibility that, for any
$\unicode[STIX]{x1D716}>0$
, there exist (large) even
$k$
such that all characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
of order
$k$
satisfy
$M(\unicode[STIX]{x1D712})\leqslant \unicode[STIX]{x1D716}\sqrt{q}\log \log q$
. We resolve this matter, by exhibiting values of
$M(\unicode[STIX]{x1D712})$
for even and odd characters
$\unicode[STIX]{x1D712}$
of even order
$k$
that are as large as the conjectured bounds (1.7). Our result also allows for the order
$k$
to grow uniformly as large as any fixed power of the logarithm of the conductor of
$\unicode[STIX]{x1D712}$
.
Theorem 1.4. Let
$A\geqslant 1$
be fixed and
$Q$
be large. Let
$2\leqslant k\leqslant (\log Q)^{A}$
be an even integer. Then there are at least
$Q^{1/3+o(1)}$
odd primitive characters
$\unicode[STIX]{x1D712}$
of order
$k$
and conductor
$q\leqslant Q$
such that
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\geqslant \biggl(\frac{\text{e}^{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D70B}}+o(1)\biggr)\sqrt{Q}\log \log Q.\end{eqnarray}$$
Moreover, there exist at least
$Q^{1/3+o(1)}$
even primitive characters
$\unicode[STIX]{x1D712}$
of order
$k$
and conductor
$q\leqslant Q$
such that
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\geqslant \biggl(\frac{\text{e}^{\unicode[STIX]{x1D6FE}}}{\unicode[STIX]{x1D70B}\sqrt{3}}+o(1)\biggr)\sqrt{Q}\log \log Q.\end{eqnarray}$$
In order to establish this result, we relate
$M(\unicode[STIX]{x1D712})$
to
$L(1,\unicode[STIX]{x1D712})$
via the following bounds, which are valid if the order of
$\unicode[STIX]{x1D712}$
is even
$$\begin{eqnarray}M(\unicode[STIX]{x1D712})\geqslant \left\{\begin{array}{@{}ll@{}}\displaystyle \frac{\sqrt{q}}{\unicode[STIX]{x1D70B}}|L(1,\unicode[STIX]{x1D712})|\quad & \text{if }\unicode[STIX]{x1D712}\text{ is odd, and has even order},\\ \displaystyle \frac{\sqrt{3q}}{2\unicode[STIX]{x1D70B}}\biggl|L\biggl(1,\unicode[STIX]{x1D712}\biggl(\frac{\cdot }{3}\biggr)\biggr)\biggr|\quad & \text{ if }\unicode[STIX]{x1D712}\text{ is even, and has even order},\\ \quad \end{array}\right.\end{eqnarray}$$
where
$(\frac{\cdot }{3})$
is the Legendre symbol modulo three. When
$\unicode[STIX]{x1D712}$
is even, this bound is proved in [Reference Bober, Goldmakher, Granville and Koukoulopoulos5, §4]. When
$\unicode[STIX]{x1D712}$
is odd, the corresponding bound follows from the pointwise estimate (see, for example, [Reference Montgomery and Vaughan19, Theorem 9.21])
$$\begin{eqnarray}\mathop{\sum }_{n\leqslant q/2}\unicode[STIX]{x1D712}(n)=(2-\unicode[STIX]{x1D712}(2))\frac{\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D712})}{\text{i}\unicode[STIX]{x1D70B}}\overline{L(1,\unicode[STIX]{x1D712})},\end{eqnarray}$$
where
$\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D712})$
is the Gauss sum associated to
$\unicode[STIX]{x1D712}$
, which satisfies
$|\unicode[STIX]{x1D70F}(\unicode[STIX]{x1D712})|=\sqrt{q}$
.
Now to obtain Theorem 1.4, it remains to produce large values of
$|L(1,\unicode[STIX]{x1D712})|$
for odd primitive characters
$\unicode[STIX]{x1D712}$
of even order
$k$
as well as large values of
$|L(1,\unicode[STIX]{x1D712}(\frac{\cdot }{3}))|$
for even primitive characters
$\unicode[STIX]{x1D712}$
of order
$k$
. Unfortunately, our construction in Theorem 1.2 does not permit us to restrict to even or odd characters, to twist by
$(\frac{\cdot }{3})$
or to allow the order
$k$
to be large. Instead, we use a different construction based on twisting a family of quadratic characters by a single specific character of order
$k$
, so that these twists also have order
$k$
, since
$k$
is even. We prove the following theorem.
Theorem 1.5. Let
$A\geqslant 1$
be fixed and
$Q$
be large. Let
$2\leqslant k\leqslant (\log Q)^{A}$
be an even integer and
$\unicode[STIX]{x1D6FF}\in \{1,-1\}$
. Let
$\unicode[STIX]{x1D709}$
be a primitive real character of conductor
$\ell$
, where
$\ell$
is fixed. Then there are at least
$Q^{1/3+o(1)}$
primitive characters
$\unicode[STIX]{x1D712}$
of order
$k$
and conductor
$q\leqslant Q$
such that
$\unicode[STIX]{x1D712}(-1)=\unicode[STIX]{x1D6FF}$
and
$$\begin{eqnarray}|L(1,\unicode[STIX]{x1D712}\unicode[STIX]{x1D709})|\geqslant \frac{\unicode[STIX]{x1D711}(\ell )}{\ell }\text{e}^{\unicode[STIX]{x1D6FE}}\log \log Q+O_{\ell ,A}(1).\end{eqnarray}$$
Using the bounds (1.8), we deduce Theorem 1.4 from Theorem 1.5 by taking
$\unicode[STIX]{x1D709}$
to be the principal character if
$\unicode[STIX]{x1D712}$
is odd and
$\unicode[STIX]{x1D709}=(\frac{\cdot }{3})$
if
$\unicode[STIX]{x1D712}$
is even.
2 Large values of
$|L(1,\unicode[STIX]{x1D712})|$
for characters
$\unicode[STIX]{x1D712}$
of order
$k$
: Proof of Theorem 1.2
Our construction in Theorem 1.2 relies on primitive characters
$\unicode[STIX]{x1D712}$
of prime conductor
$q$
. In this case, if
$\unicode[STIX]{x1D712}$
has order
$k$
, then
$q\equiv 1~(\text{mod}\;k)$
. We first prove that there are exactly
$\unicode[STIX]{x1D711}(k)$
such characters modulo
$q$
. Here and throughout, we let
$\unicode[STIX]{x1D701}_{k}:=\exp (2\unicode[STIX]{x1D70B}\text{i}/k)$
.
Lemma 2.1. Let
$k\geqslant 2$
be an integer. For any prime
$q\equiv 1\,(\text{mod}\;k)$
, there are exactly
$\unicode[STIX]{x1D711}(k)$
primitive characters of order
$k$
and conductor
$q$
.
Proof. Let
$g$
be a primitive root modulo
$q$
. Since any primitive character is completely determined by its value at
$g$
, we deduce that if
$\unicode[STIX]{x1D712}$
is primitive and
$$\begin{eqnarray}\unicode[STIX]{x1D712}(g)=(\unicode[STIX]{x1D701}_{k})^{\unicode[STIX]{x1D6FC}}\quad \text{for some }(\unicode[STIX]{x1D6FC},k)=1,\end{eqnarray}$$
then
$\unicode[STIX]{x1D712}$
has order
$k$
. Moreover, any character
$\unicode[STIX]{x1D712}$
of order
$k$
and conductor
$q$
has to satisfy (2.1). Finally, note that there are exactly
$\unicode[STIX]{x1D711}(k)$
such characters.☐
For any prime
$q\equiv 1~(\text{mod}\;k)$
, we let
$g_{q}$
be the smallest primitive root modulo
$q$
. We define
$\unicode[STIX]{x1D713}_{q}$
to be the primitive character modulo
$q$
such that
$$\begin{eqnarray}\unicode[STIX]{x1D713}_{q}(g_{q})=\unicode[STIX]{x1D701}_{k}.\end{eqnarray}$$
Note that
$\unicode[STIX]{x1D713}_{q}$
has order
$k$
. Using these characters, we are going to construct a family of primitive characters of order
$k$
, which shall be used in the proof of Theorem 1.2.
Lemma 2.2. Let
$k\geqslant 2$
be an integer and let
$m=q_{1}q_{2}$
, where
$q_{1}\neq q_{2}$
are primes such that
$q_{1}\equiv q_{2}\equiv 1\,(\text{mod}\;k)$
. Then
$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D713}}_{m}:=\unicode[STIX]{x1D713}_{q_{1}}\overline{\unicode[STIX]{x1D713}_{q_{2}}}\end{eqnarray}$$
is a primitive character of order
$k$
and conductor
$m$
.
Proof. Since
$q_{1}$
and
$q_{2}$
are coprime,
$\widetilde{\unicode[STIX]{x1D713}}_{m}$
is primitive and has conductor
$m$
. Moreover,
$\unicode[STIX]{x1D713}_{q_{1}}$
and
$\unicode[STIX]{x1D713}_{q_{2}}$
have order
$k$
and hence the order of
$\widetilde{\unicode[STIX]{x1D713}}_{m}$
divides
$k$
. Therefore, to show that
$\widetilde{\unicode[STIX]{x1D713}}_{m}$
has order
$k$
, it suffices to find a integer
$n$
such that
$\widetilde{\unicode[STIX]{x1D713}}_{m}(n)=\unicode[STIX]{x1D701}_{k}$
. Now let
$g$
be the smallest primitive root modulo
$q_{1}$
and let
$a$
be a solution to the linear congruence
$$\begin{eqnarray}aq_{2}\equiv g-1~(\text{mod}~q_{1}).\end{eqnarray}$$
Letting
$n=aq_{2}+1$
and using (2.2), we derive
$$\begin{eqnarray}\widetilde{\unicode[STIX]{x1D713}}_{m}(n)=\unicode[STIX]{x1D713}_{q_{1}}(g)\overline{\unicode[STIX]{x1D713}_{q_{2}}(aq_{2}+1)}=\unicode[STIX]{x1D701}_{k},\end{eqnarray}$$
as desired. ☐
Let
${\mathcal{F}}_{k}(Q)$
be the set of characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}$
indexed over the integers
$m=q_{1}q_{2}$
, where
$q_{1},q_{2}$
are primes such that
$\sqrt{Q}<q_{1}<q_{2}<2\sqrt{Q}$
and
$q_{1}\equiv q_{2}\equiv 1\,(\text{mod}\;k)$
. If
$A\geqslant 1$
is fixed and
$k\leqslant (\log Q)^{A}$
, then it follows, from the Siegel–Walfisz Theorem, that
$$\begin{eqnarray}\unicode[STIX]{x1D70B}(2\sqrt{Q};k,1)-\unicode[STIX]{x1D70B}(\sqrt{Q};k,1)\sim \frac{2\sqrt{Q}}{\unicode[STIX]{x1D711}(k)\log Q},\end{eqnarray}$$
and hence
$$\begin{eqnarray}|{\mathcal{F}}_{k}(Q)|\sim \frac{2Q}{(\unicode[STIX]{x1D711}(k)\log Q)^{2}}\gg \frac{Q}{(\log Q)^{2+2A}}.\end{eqnarray}$$
In order to obtain large values of
$L(1,\unicode[STIX]{x1D712})$
, a general strategy is to construct characters
$\unicode[STIX]{x1D712}$
such that
$\unicode[STIX]{x1D712}(p)=1$
for all the small primes
$p$
, typically, up to the logarithm of the conductor of
$\unicode[STIX]{x1D712}$
. Using a judicious application of the pigeonhole principle, we prove that there exist many characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}$
in
${\mathcal{F}}_{k}(Q)$
with this property.
Lemma 2.3. Let
$A\geqslant 1$
be fixed, and
$Q$
be large. Let
$2\leqslant k\leqslant (\log Q)^{A}$
be an integer and
$2\leqslant y\leqslant \log Q$
be a real number. Then there are
$$\begin{eqnarray}\gg \!\frac{Q}{k^{2\unicode[STIX]{x1D70B}(y)+2}(\log Q)^{2}}\end{eqnarray}$$
characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
such that
$\widetilde{\unicode[STIX]{x1D713}}_{m}(p)=1$
for all primes
$p\leqslant y.$
Proof. Let
$p_{j}$
denote the
$j$
th prime and put
$\ell =\unicode[STIX]{x1D70B}(y)$
. Let
$$\begin{eqnarray}U_{k}=\{\unicode[STIX]{x1D701}_{k}^{a}:0\leqslant a\leqslant k-1\}\end{eqnarray}$$
be the set of
$k$
th roots of unity and let
$$\begin{eqnarray}{\mathcal{A}}=\{\mathbf{v}=(v_{1},\ldots ,v_{\ell })\text{ such that }v_{j}\in U_{k}\text{ for all }1\leqslant j\leqslant \ell \}.\end{eqnarray}$$
Note that
$|{\mathcal{A}}|=k^{\ell }.$
For any
$\mathbf{v}\in {\mathcal{A}}$
, we define
$S_{Q}(\mathbf{v})$
to be the set of primes
$q$
in
$(\sqrt{Q},2\sqrt{Q})$
such that
$q\equiv 1~(\hspace{0.2em}mod\hspace{0.2em}~k)$
and
$\unicode[STIX]{x1D713}_{q}(p_{j})=v_{j}$
for all
$1\leqslant j\leqslant \ell$
. By (2.3),
$$\begin{eqnarray}\max _{\mathbf{v}\in {\mathcal{A}}}|S_{Q}(\mathbf{v})|\geqslant \frac{1}{|{\mathcal{A}}|}\mathop{\sum }_{\mathbf{v}\in {\mathcal{A}}}|S_{Q}(\mathbf{v})|\gg \frac{\sqrt{Q}}{k^{\ell }\unicode[STIX]{x1D711}(k)\log Q}.\end{eqnarray}$$
Let
$\mathbf{v}_{\text{max}}$
be such that
$\max _{\mathbf{v}\in {\mathcal{A}}}|S_{Q}(\mathbf{v})|=|S_{Q}(\mathbf{v}_{\text{max}})|.$
Note that if
$q_{1}\neq q_{2}$
are both in
$S_{Q}(\mathbf{v}_{\text{max}})$
, then
$\unicode[STIX]{x1D713}_{q_{1}}\overline{\unicode[STIX]{x1D713}_{q_{2}}}(p_{j})=1$
for all
$1\leqslant j\leqslant \ell .$
Finally, the number of characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
such that
$m=q_{1}q_{2}$
and
$q_{1},q_{2}\in S_{Q}(\mathbf{v}_{\text{max}})$
equals
$$\begin{eqnarray}\binom{|S_{Q}(\mathbf{v}_{\text{max}})|}{2}\gg \frac{Q}{k^{2\ell +2}(\log Q)^{2}},\end{eqnarray}$$
as desired. ☐
The second ingredient in the proof of Theorem 1.2 is to approximate
$L(1,\unicode[STIX]{x1D712})$
by a short Euler product. Using zero density estimates together with the large sieve, Granville and Soundararajan [Reference Granville and Soundararajan12] proved that this can be done for almost all primitive characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
with
$q\leqslant Q$
. More precisely, they established the following proposition.
Proposition 2.4 [Reference Granville and Soundararajan12, Proposition 2.2].
Let
$A>2$
be fixed. Then, for all but at most
$Q^{2/A+o(1)}$
primitive characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
with
$q\leqslant Q$
,
$$\begin{eqnarray}L(1,\unicode[STIX]{x1D712})=\mathop{\prod }_{p\leqslant (\log Q)^{A}}\biggl(1-\frac{\unicode[STIX]{x1D712}(p)}{p}\biggr)^{-1}\biggl(1+O\biggl(\frac{1}{\log \log Q}\biggr)\biggr).\end{eqnarray}$$
Let
$z=(\log Q)^{A}$
for some
$A>2$
and let
$y\leqslant (\log Q)$
be a large real number. Then, note that
$$\begin{eqnarray}\displaystyle \mathop{\prod }_{p\leqslant z}\biggl(1-\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr)^{-1} & = & \displaystyle \mathop{\prod }_{p\leqslant y}\biggl(1-\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr)^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \exp \biggl(\mathop{\sum }_{y<p<z}\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}+O\biggl(\frac{1}{\sqrt{y}\log y}\biggr)\biggr).\nonumber\end{eqnarray}$$
Let
$k\geqslant 2$
be a fixed integer. By Lemma 2.3, there are at least
$Q^{1+o(1)}$
characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
for which the product
$\prod _{p\leqslant y}(1-\widetilde{\unicode[STIX]{x1D713}}_{m}(p)/p)^{-1}$
is as large as possible. The third and last ingredient in the proof of Theorem 1.2 is the following proposition that gives an upper bound for the
$2r$
th moment of
$\sum _{y<p<z}\widetilde{\unicode[STIX]{x1D713}}_{m}(p)/p$
over the characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
uniformly for
$r$
in a large range. In particular, we shall later deduce that, with very few exceptions in
${\mathcal{F}}_{k}(Q)$
, the prime sum
$\sum _{y<p<z}\widetilde{\unicode[STIX]{x1D713}}_{m}(p)/p$
is small.
Proposition 2.5. Let
$k\geqslant 2$
be a fixed integer and let
$k<y<(\log Q)^{A}$
be a real number, where
$A>2$
is a constant. Put
$z=(\log Q)^{A}$
. Then, for every positive integer
$r\leqslant \log Q/(3Ak^{2}\log \log Q)$
,
$$\begin{eqnarray}\mathop{\sum }_{\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)}\biggl|\mathop{\sum }_{y<p<z}\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr|^{2r}\ll _{k}2^{r}r!Q\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{2}}\biggr)^{r}+Q^{1-1/(4k)}.\end{eqnarray}$$
To establish this result, we shall use the following large sieve inequality over
$k$
th order characters, which is due to Elliott [Reference Elliott7].
Lemma 2.6 [Reference Elliott7, Lemma 33].
Let
$k\geqslant 2$
be a fixed integer and let
$\{\unicode[STIX]{x1D706}_{n}\}_{n\geqslant 1}$
be a sequence of complex numbers. Then
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ p\leqslant Q \\ p\equiv 1~(\text{mod}~k)}}~~\sum ^{\star }_{\unicode[STIX]{x1D712}\,(\text{mod}\,p)}\biggl|\mathop{\sum }_{n\leqslant H}\unicode[STIX]{x1D706}_{n}\unicode[STIX]{x1D712}(n)\biggr|^{2}\nonumber\\ \displaystyle & & \displaystyle \qquad \ll _{k}~Q\biggl|\mathop{\sum }_{\substack{ m,n\leqslant H \\ mn^{k-1}=\mathfrak{a}^{k}}}\unicode[STIX]{x1D706}_{n}\overline{\unicode[STIX]{x1D706}_{m}}\biggr|+H^{k}Q^{1-2/(k+1)}\biggl(\mathop{\sum }_{n\leqslant H}|\unicode[STIX]{x1D706}_{n}|\biggr)^{2},\nonumber\end{eqnarray}$$
where the sum
${\sum ^{\star }}_{\unicode[STIX]{x1D712}\,(\text{mod}\;p)}$
is confined to characters of order
$k$
modulo
$p$
for each prime modulus
$p$
, and
$\mathfrak{a}$
is an algebraic integer of the cyclotomic field
$K=\mathbb{Q}(\unicode[STIX]{x1D701}_{k})$
.
Remark 2.7. If the sequence
$\unicode[STIX]{x1D706}_{n}$
is supported only on integers
$n$
which are coprime to
$k$
(that is, if
$\unicode[STIX]{x1D706}_{n}=0$
for all
$(n,k)>1$
), then one might replace the condition
$mn^{k-1}=\mathfrak{a}^{k}$
by
$mn^{k-1}$
is an integral
$k$
th power (that is, one might take
$\mathfrak{a}\in \mathbb{N}$
). To see this, put
$M=mn^{k-1}$
and let
$M=p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{\ell }^{\unicode[STIX]{x1D6FC}_{\ell }}$
be its prime factorization in
$\mathbb{N}$
. If
$M=\mathfrak{a}^{k}$
for some algebraic integer
$\mathfrak{a}$
of
$K$
, then
$p_{j}^{\unicode[STIX]{x1D6FC}_{j}}=\mathfrak{a}_{j}^{k}$
for some algebraic integers
$\mathfrak{a}_{1},\ldots ,\mathfrak{a}_{\ell }$
of
$K$
. Moreover, since
$p_{j}\nmid k$
,
$p_{j}$
is unramified in
$K$
, and hence we must have
$k\mid \unicode[STIX]{x1D6FC}_{j}$
for all
$1\leqslant j\leqslant \ell$
, which implies that
$M$
is an integral
$k$
th power.
Proof of Proposition 2.5.
First, observe that
$$\begin{eqnarray}\biggl(\mathop{\sum }_{y<p<z}\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr)^{r}=\mathop{\sum }_{y^{r}<n<z^{r}}\frac{b_{r}(n)\widetilde{\unicode[STIX]{x1D713}}_{m}(n)}{n},\end{eqnarray}$$
where
$$\begin{eqnarray}b_{r}(n):=\mathop{\sum }_{\substack{ y<p_{1},\ldots ,p_{r}<z \\ p_{1}\cdots p_{r}=n}}1.\end{eqnarray}$$
Note that
$0\leqslant b_{r}(n)\leqslant r!$
. Moreover,
$b_{r}(n)=0$
unless
$n=p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{s}^{\unicode[STIX]{x1D6FC}_{s}}$
where
$y<p_{1}<p_{2}<\cdots <p_{s}<z$
are distinct primes and
$\unicode[STIX]{x1D6FA}(n)=\unicode[STIX]{x1D6FC}_{1}+\cdots +\unicode[STIX]{x1D6FC}_{s}=r$
(where
$\unicode[STIX]{x1D6FA}(n)$
is the number of prime divisors of
$n$
counting multiplicities). In this case,
$$\begin{eqnarray}b_{r}(n)=\binom{r}{\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{s}}.\end{eqnarray}$$
Using this formula, one can easily deduce that if
$n_{1}$
and
$n_{2}$
are positive integers with
$\unicode[STIX]{x1D6FA}(n_{1})=r_{1}$
and
$\unicode[STIX]{x1D6FA}(n_{2})=r_{2}$
, then
$$\begin{eqnarray}b_{r_{1}+r_{2}}(n_{1}n_{2})\leqslant \binom{r_{1}+r_{2}}{r_{1}}b_{r_{1}}(n_{1})b_{r_{2}}(n_{2}).\end{eqnarray}$$
Recall that, for all
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
, there exist primes
$\sqrt{Q}<q_{1}<q_{2}<2\sqrt{Q}$
such that
$q_{1}\equiv q_{2}\equiv 1\,(\text{mod}\;k)$
and
$m=q_{1}q_{2}$
. In this case,
$\widetilde{\unicode[STIX]{x1D713}}_{m}(n)=\unicode[STIX]{x1D713}_{q_{1}}(n)\overline{\unicode[STIX]{x1D713}_{q_{2}}}(n).$
Therefore,
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)}\biggl|\mathop{\sum }_{y<p<z}\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr|^{2r}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant \mathop{\sum }_{\substack{ q_{1}<2\sqrt{Q} \\ q_{1}\equiv 1\hspace{0.2em}\bmod \hspace{0.2em}k}}~\mathop{\sum }_{\substack{ q_{2}<2\sqrt{Q} \\ q_{2}\equiv 1\hspace{0.2em}\bmod \hspace{0.2em}k}}\biggl|\mathop{\sum }_{y^{r}<n<z^{r}}\frac{b_{r}(n)\overline{\unicode[STIX]{x1D713}_{q_{1}}(n)}}{n}\unicode[STIX]{x1D713}_{q_{2}}(n)\biggr|^{2}.\end{eqnarray}$$
We define
$$\begin{eqnarray}\unicode[STIX]{x1D706}_{n}=\frac{b_{r}(n)\overline{\unicode[STIX]{x1D713}_{q_{1}}(n)}}{n}.\end{eqnarray}$$
Since
$y>k$
,
$b_{r}(n)=0$
if
$(n,k)>1$
. Therefore, it follows, from Lemma 2.6 and Remark 2.7, that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ q_{2}<2\sqrt{Q} \\ q_{2}\equiv 1\hspace{0.2em}\bmod \hspace{0.2em}k}}\biggl|\mathop{\sum }_{y^{r}<n<z^{r}}\frac{b_{r}(n)\overline{\unicode[STIX]{x1D713}_{q_{1}}(n)}}{n}\unicode[STIX]{x1D713}_{q_{2}}(n)\biggr|^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \ll _{k}~Q^{1/2}\biggl|\mathop{\sum }_{\substack{ y^{r}<n_{1},n_{2}<z^{r} \\ n_{1}n_{2}^{k-1}\text{ is a }k\text{th power}}}\frac{b_{r}(n_{1})b_{r}(n_{2})\overline{\unicode[STIX]{x1D713}_{q_{1}}(n_{1})}\unicode[STIX]{x1D713}_{q_{1}}(n_{2})}{n_{1}n_{2}}\biggr|+E_{1}(Q),\nonumber\\ \displaystyle & & \displaystyle \quad \ll _{k}~Q^{1/2}\mathop{\sum }_{\substack{ y^{r}<n_{1},n_{2}<z^{r} \\ n_{1}n_{2}^{k-1}\text{ is a }k\text{th power}}}\frac{b_{r}(n_{1})b_{r}(n_{2})}{n_{1}n_{2}}+E_{1}(Q),\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle E_{1}(Q) & \ll _{k} & \displaystyle z^{rk}Q^{1/2-1/(k+1)}\biggl(\mathop{\sum }_{y^{r}\leqslant n\leqslant z^{r}}\frac{b_{r}(n)}{n}\biggr)^{2}\nonumber\\ \displaystyle & \ll & \displaystyle z^{rk}Q^{1/2-1/(k+1)}\biggl(\mathop{\sum }_{y\leqslant p\leqslant z}\frac{1}{p}\biggr)^{2r}\ll Q^{1/2-1/(4k)}.\end{eqnarray}$$
Next, we bound the main term on the right-hand side of (2.8). Let
$n_{1}$
and
$n_{2}$
be positive integers such that
$\unicode[STIX]{x1D6FA}(n_{1})=\unicode[STIX]{x1D6FA}(n_{2})=r$
and put
$d=(n_{1},n_{2})$
. Also, put
$n_{1}=dm_{1}$
and
$n_{2}=dm_{2}$
. Since
$n_{1}n_{2}^{k-1}$
is a
$k$
th power, both
$m_{1}$
and
$m_{2}$
are
$k$
th powers, since
$(m_{1},m_{2})=1.$
Let
$m_{1}=\ell _{1}^{k}$
and
$m_{2}=\ell _{2}^{k}$
and put
$s=\unicode[STIX]{x1D6FA}(\ell _{1})$
. Since
$\unicode[STIX]{x1D6FA}(n_{1})=\unicode[STIX]{x1D6FA}(n_{2})=r$
,
$\unicode[STIX]{x1D6FA}(\ell _{2})=s$
and
$\unicode[STIX]{x1D6FA}(d)=r-ks$
. Therefore, by (2.6), we obtain
$$\begin{eqnarray}\displaystyle b_{r}(n_{1})b_{r}(n_{2}) & {\leqslant} & \displaystyle \binom{r}{ks}^{2}b_{ks}(\ell _{1}^{k})b_{ks}(\ell _{2}^{k})b_{r-ks}(d)^{2}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{(r!)^{2}}{(ks)!^{2}(r-ks)!}b_{ks}(\ell _{1}^{k})b_{ks}(\ell _{2}^{k})b_{r-ks}(d),\end{eqnarray}$$
since
$b_{t}(n)\leqslant t!$
for any positive integers
$t$
and
$n$
. Furthermore, by (2.6) together with a simple inductive argument, we derive
$$\begin{eqnarray}b_{ks}(\ell _{i}^{k})\leqslant \frac{(ks)!}{(s!)^{k}}b_{s}(\ell _{i})^{k}\leqslant \frac{(ks)!}{s!}b_{s}(\ell _{i})\quad \text{for }i=1,2.\end{eqnarray}$$
Inserting this estimate in (2.10) yields
$$\begin{eqnarray}\displaystyle b_{r}(n_{1})b_{r}(n_{2}) & {\leqslant} & \displaystyle \frac{(r!)^{2}}{(s!)^{2}(r-ks)!}b_{s}(\ell _{1})b_{s}(\ell _{2})b_{r-ks}(d)\nonumber\\ \displaystyle & {\leqslant} & \displaystyle r!\binom{r}{ks}b_{s}(\ell _{1})b_{s}(\ell _{2})b_{r-ks}(d),\nonumber\end{eqnarray}$$
since
$(ks)!\geqslant (2s)!\geqslant (s!)^{2}.$
Thus we deduce that
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ y^{r}<n_{1},n_{2}<z^{r} \\ n_{1}n_{2}^{k-1}\text{ is a }k\text{th power}}}\frac{b_{r}(n_{1})b_{r}(n_{2})}{n_{1}n_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad \leqslant r!\mathop{\sum }_{0\leqslant s\leqslant r/k}\binom{r}{ks}\biggl(\mathop{\sum }_{d}\frac{b_{r-ks}(d)}{d^{2}}\biggr)\biggl(\mathop{\sum }_{\ell }\frac{b_{s}(\ell )}{\ell ^{k}}\biggr)^{2}\nonumber\\ \displaystyle & & \displaystyle \quad =r!\mathop{\sum }_{0\leqslant s\leqslant r/k}\binom{r}{sk}\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{2}}\biggr)^{r-sk}\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{k}}\biggr)^{2s},\end{eqnarray}$$
since
$$\begin{eqnarray}\mathop{\sum }_{n}\frac{b_{t}(n)}{n^{\unicode[STIX]{x1D6FC}}}=\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{\unicode[STIX]{x1D6FC}}}\biggr)^{t}.\end{eqnarray}$$
Furthermore, since
$k\geqslant 2$
, for any positive integer
$n$
, the Euclidean norm in
$\mathbb{R}^{n}$
is larger than the
$k$
-norm. Therefore,
$$\begin{eqnarray}\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{k}}\biggr)^{1/k}\leqslant \biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{2}}\biggr)^{1/2}.\end{eqnarray}$$
Inserting this bound in (2.11) yields
$$\begin{eqnarray}\mathop{\sum }_{\substack{ y^{r}<n_{1},n_{2}<z^{r} \\ n_{1}n_{2}^{k-1}\text{ is a }k\text{th power }}}\frac{b_{r}(n_{1})b_{r}(n_{2})}{n_{1}n_{2}}\leqslant 2^{r}r!\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{2}}\biggr)^{r}.\end{eqnarray}$$
Combining this bound with equations (2.7), (2.8) and (2.9) completes the proof. ☐
We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2.
Let
$z=(\log Q)^{5}$
and let
$2\leqslant y\leqslant (\log Q)^{3/2}$
be a real number to be chosen later. Then, by Proposition 2.4, it follows that for all but at most
$Q^{1/2}$
primitive characters
$\unicode[STIX]{x1D712}\,(\text{mod}\;q)$
with
$q\leqslant Q$
,
$$\begin{eqnarray}\displaystyle L(1,\unicode[STIX]{x1D712}) & = & \displaystyle \mathop{\prod }_{p\leqslant z}\biggl(1-\frac{\unicode[STIX]{x1D712}(p)}{p}\biggr)^{-1}\biggl(1+O\biggl(\frac{1}{\log \log Q}\biggr)\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\prod }_{p\leqslant y}\biggl(1-\frac{\unicode[STIX]{x1D712}(p)}{p}\biggr)^{-1}\exp \biggl(\mathop{\sum }_{y<p<z}\frac{\unicode[STIX]{x1D712}(p)}{p}\biggr)\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl(1+O\biggl(\frac{1}{\log \log Q}+\frac{1}{\sqrt{y}\log y}\biggr)\biggr),\end{eqnarray}$$
by the prime number theorem.
Furthermore, by taking
$r=[\log Q/(15k^{2}\log \log Q)]$
in Proposition 2.5, we obtain that the number of characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
such that
$$\begin{eqnarray}\biggl|\mathop{\sum }_{y<p<z}\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr|>\frac{1}{\log \log Q}\end{eqnarray}$$
is
$$\begin{eqnarray}\ll Q\biggl(\frac{4r(\log \log Q)^{2}}{y\log y}\biggr)^{r}\ll Q\biggl(\frac{\log Q\log \log Q}{3k^{2}y\log y}\biggr)^{r},\end{eqnarray}$$
since
$\sum _{y<p}1/p^{2}\leqslant 2/y\log y$
, by the prime number theorem.
On the other hand, it follows, from Lemma 2.3, that, for some constant
$c_{k}>0$
, there are at least
$Q\exp (-c_{k}\log Q/\text{log}\log Q)$
characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
for which
$\widetilde{\unicode[STIX]{x1D713}}_{m}(p)=1$
for all primes
$p\leqslant \log Q$
. Choosing
$y=b_{k}\log Q$
for some suitably large constant
$b_{k}>0$
, we deduce, from (2.12) and (2.13), that there are at least
$Q\exp (-2c_{k}\log Q/\text{log}\log Q)$
characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}\in {\mathcal{F}}_{k}(Q)$
for which
$\widetilde{\unicode[STIX]{x1D713}}_{m}(p)=1$
for all primes
$p\leqslant \log Q$
such that (2.12) holds and where
$$\begin{eqnarray}\biggl|\mathop{\sum }_{y<p<z}\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr|\leqslant \frac{1}{\log \log Q}.\end{eqnarray}$$
For these characters
$\widetilde{\unicode[STIX]{x1D713}}_{m}$
, by (2.12),
$$\begin{eqnarray}\displaystyle L(1,\widetilde{\unicode[STIX]{x1D713}}_{m}) & = & \displaystyle \mathop{\prod }_{p\leqslant \log Q}\biggl(1-\frac{1}{p}\biggr)^{-1}\exp \biggl(\mathop{\sum }_{ logQ<p<y}\frac{\widetilde{\unicode[STIX]{x1D713}}_{m}(p)}{p}\biggr)\biggl(1+O\biggl(\frac{1}{\log \log Q}\biggr)\biggr)\nonumber\\ \displaystyle & = & \displaystyle \text{e}^{\unicode[STIX]{x1D6FE}}\log \log Q+O_{k}(1),\nonumber\end{eqnarray}$$
as desired. ☐
3 Large values of
$|L(1,\unicode[STIX]{x1D712}\unicode[STIX]{x1D709})|$
for even-order characters
$\unicode[STIX]{x1D712}$
: proof of Theorem 1.5
It follows, from Lemma 2.3, that there is a character
$\unicode[STIX]{x1D713}$
of large conductor
$m=q_{1}q_{2}$
, where
$q_{1}$
and
$q_{2}$
are primes, such that
$\unicode[STIX]{x1D713}$
has order
$k\leqslant (\log m)^{A}$
and
$\unicode[STIX]{x1D713}(p)=1$
for all primes
$p\leqslant (\log m)/(4A)$
. In order to prove Theorem 1.5, our construction involves the characters
$\unicode[STIX]{x1D713}\unicode[STIX]{x1D712}_{d}$
, where
$d$
ranges over a certain family of fundamental discriminants. To exhibit large values of
$L(1,\unicode[STIX]{x1D713}\unicode[STIX]{x1D712}_{d}\unicode[STIX]{x1D709})$
, we first prove that, for many of these fundamental discriminants
$d$
,
$\unicode[STIX]{x1D712}_{d}(p)=\unicode[STIX]{x1D709}(p)$
for all the “small” primes
$p$
.
Proposition 3.1. Let
$\unicode[STIX]{x1D6FF}=\pm 1$
, let
$Q$
be large and let
$2\leqslant y\leqslant \log Q$
be a real number. Put
$P(y)=\prod _{p\leqslant y}p$
. Let
$\unicode[STIX]{x1D700}(p)=\pm 1$
for each prime
$p$
. Then there are
$$\begin{eqnarray}\gg \frac{Q}{2^{\unicode[STIX]{x1D70B}(y)}\log y}\end{eqnarray}$$
fundamental discriminants
$d\equiv 1~(\text{mod}~4)$
such that
$0<\unicode[STIX]{x1D6FF}d\leqslant Q$
,
$(d,P(y))=1$
and
$\unicode[STIX]{x1D712}_{d}(p)=\unicode[STIX]{x1D700}(p)$
for all primes
$p\leqslant y$
.
To establish this proposition, we first need to count the fundamental discriminants
$d$
such that
$0<\unicode[STIX]{x1D6FF}d\leqslant Q$
,
$d\equiv 1~(\text{mod}~4)$
and
$p\mid d\;\Longrightarrow \;p>y$
. To this end, we use the following standard estimate whose proof we include for completeness.
Lemma 3.2. Let
$m$
be a positive integer and
$Q$
be a large real number. The number of fundamental discriminants
$d$
such that
$0<\unicode[STIX]{x1D6FF}d\leqslant Q$
,
$d\equiv 1~(\text{mod}~4)$
and
$(d,m)=1$
equals
$$\begin{eqnarray}\frac{3}{\unicode[STIX]{x1D70B}^{2}}Q\mathop{\prod }_{p|2m}\biggl(1+\frac{1}{p}\biggr)^{-1}+O(d(m)Q^{1/2}),\end{eqnarray}$$
where
$d(m)=\sum _{b|m}1$
is the divisor function.
Proof. We only prove the estimate when
$\unicode[STIX]{x1D6FF}=1$
, since the proof for the case
$\unicode[STIX]{x1D6FF}=-1$
follows along similar lines. Let
$\unicode[STIX]{x1D712}_{-4}$
be the non-principal real primitive character modulo four and let
$N_{m}(Q)$
be the number of fundamental discriminants
$0<d\leqslant Q$
such
$d\equiv 1~(\text{mod}~4)$
and
$(d,m)=1$
. Then, writing
$\unicode[STIX]{x1D707}^{2}(d)=\sum _{h^{2}\mid d}\unicode[STIX]{x1D707}(h)$
, we obtain
$$\begin{eqnarray}\displaystyle N_{m}(Q) & = & \displaystyle \mathop{\sum }_{\substack{ 0<d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,m)=1}}\unicode[STIX]{x1D707}^{2}(d)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\mathop{\sum }_{\substack{ 0<d\leqslant Q \\ (d,2m)=1}}(1+\unicode[STIX]{x1D712}_{-4}(d))\mathop{\sum }_{h^{2}|d}\unicode[STIX]{x1D707}(h)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\mathop{\sum }_{\substack{ h\leqslant \sqrt{Q} \\ (h,2m)=1}}\unicode[STIX]{x1D707}(h)\mathop{\sum }_{\substack{ b\leqslant Q/h^{2} \\ (b,2m)=1}}(1+\unicode[STIX]{x1D712}_{-4}(b)).\end{eqnarray}$$
Now observe that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ b\leqslant Q/h^{2} \\ (b,2m)=1}}1 & = & \displaystyle \mathop{\sum }_{a|2m}\unicode[STIX]{x1D707}(a)\mathop{\sum }_{\substack{ b\leqslant Q/h^{2} \\ a|b}}1=\frac{Q}{h^{2}}\mathop{\sum }_{a|2m}\frac{\unicode[STIX]{x1D707}(a)}{a}+O(d(2m))\nonumber\\ \displaystyle & = & \displaystyle \frac{Q}{h^{2}}\mathop{\prod }_{p|2m}\biggl(1-\frac{1}{p}\biggr)+O(d(2m)).\nonumber\end{eqnarray}$$
Similarly,
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ b\leqslant Q/h^{2} \\ (b,2m)=1}}\unicode[STIX]{x1D712}_{-4}(b) & = & \displaystyle \mathop{\sum }_{a|2m}\unicode[STIX]{x1D707}(a)\mathop{\sum }_{\substack{ b\leqslant Q/h^{2} \\ a|b}}\unicode[STIX]{x1D712}_{-4}(b)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{a|2m}\unicode[STIX]{x1D707}(a)\unicode[STIX]{x1D712}_{-4}(a)\mathop{\sum }_{r\leqslant Q/(ah^{2})}\unicode[STIX]{x1D712}_{-4}(r)=O(d(2m)).\nonumber\end{eqnarray}$$
Combining these estimates with (3.1) and using that
$d(2m)\leqslant 2d(m)$
, we deduce that
$$\begin{eqnarray}N_{m}(Q)=\frac{Q}{2}\mathop{\prod }_{p|2m}\biggl(1-\frac{1}{p}\biggr)\mathop{\sum }_{\substack{ h\leqslant \sqrt{Q} \\ (h,2m)=1}}\frac{\unicode[STIX]{x1D707}(h)}{h^{2}}+O(d(m)Q^{1/2}).\end{eqnarray}$$
Finally, note that
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ h\leqslant \sqrt{Q} \\ (h,2m)=1}}\frac{\unicode[STIX]{x1D707}(h)}{h^{2}} & = & \displaystyle \mathop{\sum }_{\substack{ h\geqslant 1 \\ (h,2m)=1}}\frac{\unicode[STIX]{x1D707}(h)}{h^{2}}+O(Q^{-1/2})\nonumber\\ \displaystyle & = & \displaystyle \frac{6}{\unicode[STIX]{x1D70B}^{2}}\mathop{\prod }_{p|2m}\biggl(1-\frac{1}{p^{2}}\biggr)^{-1}+O(Q^{-1/2}).\nonumber\end{eqnarray}$$
Inserting this estimate in (3.2) completes the proof. ☐
The second ingredient in the proof of Proposition 3.1 is the following bound on character sums, which is a slight variation of Lemma 4.1 of Granville and Soundararajan [Reference Granville and Soundararajan12]. Here and throughout, the sum
${\sum ^{\flat }}_{d}$
is confined to fundamental discriminants
$d$
.
Lemma 3.3. Let
$\unicode[STIX]{x1D6FF}=\pm 1$
. Let
$m$
be positive integer and let
$n\geqslant 2$
be an integer, not a perfect square. Then
$$\begin{eqnarray}\sum ^{\flat }_{\substack{ 0<\unicode[STIX]{x1D6FF}d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,m)=1}}\unicode[STIX]{x1D712}_{d}(n)\ll d(m)Q^{1/2}n^{1/4}(\log n)^{1/2}.\end{eqnarray}$$
Proof. We only prove the estimate when
$\unicode[STIX]{x1D6FF}=1$
, since the proof for the case
$\unicode[STIX]{x1D6FF}=-1$
follows similarly. Writing
$\unicode[STIX]{x1D707}^{2}(d)=\sum _{h^{2}\mid d}\unicode[STIX]{x1D707}(h)$
, we get
$$\begin{eqnarray}\displaystyle \sum ^{\flat }_{\substack{ 0<d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,m)=1}}\unicode[STIX]{x1D712}_{d}(n) & = & \displaystyle \mathop{\sum }_{\substack{ d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,m)=1}}\unicode[STIX]{x1D707}^{2}(d)\unicode[STIX]{x1D712}_{d}(n)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\mathop{\sum }_{\unicode[STIX]{x1D709}~(\text{mod}~4)}\mathop{\sum }_{\substack{ h\leqslant \sqrt{Q} \\ (h,m)=1}}\unicode[STIX]{x1D707}(h)\mathop{\sum }_{\substack{ d\leqslant Q \\ h^{2}|d \\ (d,m)=1}}\unicode[STIX]{x1D709}(d)\biggl(\frac{d}{n}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\mathop{\sum }_{\unicode[STIX]{x1D709}~(\text{mod}~4)}\mathop{\sum }_{\substack{ h\leqslant \sqrt{Q} \\ (h,m)=1}}\unicode[STIX]{x1D707}(h)\unicode[STIX]{x1D709}(h^{2})\biggl(\frac{h^{2}}{n}\biggr)\mathop{\sum }_{\substack{ b\leqslant Q/h^{2} \\ (b,m)=1}}\unicode[STIX]{x1D709}(b)\biggl(\frac{b}{n}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\mathop{\sum }_{\unicode[STIX]{x1D709}~(\text{mod}~4)}\mathop{\sum }_{\substack{ h\leqslant \sqrt{Q} \\ (h,m)=1}}\unicode[STIX]{x1D707}(h)\unicode[STIX]{x1D709}(h^{2})\biggl(\frac{h^{2}}{n}\biggr)\mathop{\sum }_{a|m}\unicode[STIX]{x1D707}(a)\mathop{\sum }_{\substack{ b\leqslant Q/h^{2} \\ a|b}}\unicode[STIX]{x1D709}(b)\biggl(\frac{b}{n}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{1}{2}\mathop{\sum }_{\unicode[STIX]{x1D709}~(\text{mod}~4)}\mathop{\sum }_{\substack{ h\leqslant \sqrt{Q} \\ (h,m)=1}}\unicode[STIX]{x1D707}(h)\unicode[STIX]{x1D709}(h^{2})\biggl(\frac{h^{2}}{n}\biggr)\mathop{\sum }_{a|m}\unicode[STIX]{x1D707}(a)\unicode[STIX]{x1D709}(a)\biggl(\frac{a}{n}\biggr)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{\ell \leqslant Q/(ah^{2})}\unicode[STIX]{x1D709}(\ell )\biggl(\frac{\ell }{n}\biggr).\end{eqnarray}$$
Since
$\unicode[STIX]{x1D709}(\cdot )(\frac{\cdot }{n})$
is a non-principal character of conductor at most
$4n$
, by the Pólya–Vinogradov inequality,
$$\begin{eqnarray}\mathop{\sum }_{\ell \leqslant Q/(ah^{2})}\unicode[STIX]{x1D709}(\ell )\biggl(\frac{\ell }{n}\biggr)\ll \sqrt{n}\log n.\end{eqnarray}$$
We use this bound in (3.3) if
$h\leqslant Q^{1/2}n^{-1/4}(\log n)^{-1/2}$
and use the trivial bound
$Q/h^{2}$
if
$h>Q^{1/2}n^{-1/4}(\log n)^{-1/2}$
in order to get
$$\begin{eqnarray}\displaystyle \sum ^{\flat }_{\substack{ 0<d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,m)=1}}\unicode[STIX]{x1D712}_{d}(n) & \ll & \displaystyle d(m)Q^{1/2}n^{1/4}(\log n)^{1/2}+d(m)Q\mathop{\sum }_{h>Q^{1/2}n^{-1/4}(\log n)^{-1/2}}\frac{1}{h^{2}}\nonumber\\ \displaystyle & \ll & \displaystyle d(m)Q^{1/2}n^{1/4}(\log n)^{1/2},\nonumber\end{eqnarray}$$
as desired. ☐
Proof of Proposition 3.1.
Let
$L=\unicode[STIX]{x1D70B}(y)$
. Note that if
$(d,P(y))=1$
and
$p$
is a prime
${\leqslant}y$
, then
$1+\unicode[STIX]{x1D712}_{d}(p)\unicode[STIX]{x1D700}(p)=2$
if
$\unicode[STIX]{x1D712}_{d}(p)=\unicode[STIX]{x1D700}(p)$
and equals zero otherwise. Therefore, the number of fundamental discriminants
$d\equiv 1~(\text{mod}~4)$
such that
$0<\unicode[STIX]{x1D6FF}d\leqslant Q$
,
$(d,P(y))=1$
and
$\unicode[STIX]{x1D712}_{d}(p)=\unicode[STIX]{x1D700}(p)$
for all primes
$p\leqslant y$
equals
$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{1}{2^{L}}\sum ^{\flat }_{\substack{ 0<\unicode[STIX]{x1D6FF}d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,P(y))=1}}~\mathop{\prod }_{p\leqslant y}(1+\unicode[STIX]{x1D712}_{d}(p)\unicode[STIX]{x1D700}(p))\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{2^{L}}\sum ^{\flat }_{\substack{ 0<\unicode[STIX]{x1D6FF}d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,P(y))=1}}\biggl(1+\mathop{\sum }_{r=1}^{L}\mathop{\sum }_{p_{1}<p_{2}<\cdots <p_{r}\leqslant y}\unicode[STIX]{x1D700}(p_{1})\cdots \unicode[STIX]{x1D700}(p_{r})\unicode[STIX]{x1D712}_{d}(p_{1}\cdots p_{r})\biggr)\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{2^{L}}\sum ^{\flat }_{\substack{ 0<\unicode[STIX]{x1D6FF}d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,P(y))=1}}1+\frac{1}{2^{L}}\mathop{\sum }_{r=1}^{L}\mathop{\sum }_{p_{1}<p_{2}<\cdots <p_{r}\leqslant y}\unicode[STIX]{x1D700}(p_{1})\cdots \unicode[STIX]{x1D700}(p_{r})\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\sum ^{\flat }_{\substack{ 0<\unicode[STIX]{x1D6FF}d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,P(y))=1}}\unicode[STIX]{x1D712}_{d}(p_{1}\cdots p_{r})\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{2^{L}}\sum ^{\flat }_{\substack{ 0<\unicode[STIX]{x1D6FF}d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,P(y))=1}}1+O(Q^{1/2}\text{e}^{y/3}),\nonumber\end{eqnarray}$$
by Lemma 3.3 together with the fact that
$\prod _{p\leqslant y}p=\text{e}^{y(1+o(1))}$
, which follows from the prime number theorem. Finally, using Lemma 3.2, we get
$$\begin{eqnarray}\sum ^{\flat }_{\substack{ 0<d\leqslant Q \\ d\equiv 1~(\text{mod}~4) \\ (d,P(y))=1}}1\gg \frac{Q}{\log y},\end{eqnarray}$$
which completes the proof. ☐
We also need an
$L^{2r}$
bound for the prime sum
$\sum _{y<p<z}(\unicode[STIX]{x1D713}\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}_{d})(p)/p$
, similar to Proposition 2.5. To this end we establish the following lemma.
Lemma 3.4. Let
$\{a(p)\}_{p\,\text{ prime}}$
be a sequence of complex numbers such that
$|a(p)|\leqslant 1$
. Let
$A\geqslant 1$
be fixed and let
$2\leqslant y<(\log Q)^{A}$
be a real number. Put
$z=(\log Q)^{A}$
. Then, for any positive integer
$r\leqslant \log Q/(6A\log \log Q)$
,
$$\begin{eqnarray}\frac{1}{Q}\sum ^{\flat }_{\substack{ |d|\leqslant Q \\ d\equiv 1~(\text{mod}~4)}}\biggl(\mathop{\sum }_{y<p<z}\frac{a(p)\unicode[STIX]{x1D712}_{d}(p)}{p}\biggr)^{2r}\ll \frac{(2r)!}{r!}\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{2}}\biggr)^{r}+Q^{-1/3}.\end{eqnarray}$$
Proof. First, we extend the sequence
$\{a(p)\}_{p}$
multiplicatively to all positive integers
$n>1$
by setting
$$\begin{eqnarray}a(n)=a(p_{1})^{\unicode[STIX]{x1D6FC}_{1}}\cdots a(p_{j})^{\unicode[STIX]{x1D6FC}_{j}}\quad \text{if }n=p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{j}^{\unicode[STIX]{x1D6FC}_{j}}.\end{eqnarray}$$
Then
$$\begin{eqnarray}\biggl(\mathop{\sum }_{y<p<z}\frac{a(p)\unicode[STIX]{x1D712}_{d}(p)}{p}\biggr)^{2r}=\mathop{\sum }_{y^{2r}<n<z^{2r}}\frac{a(n)\unicode[STIX]{x1D712}_{d}(n)b_{2r}(n)}{n},\end{eqnarray}$$
where the coefficient
$b_{2r}(n)$
is defined in (2.4). Therefore, using Lemma 3.3, we get
$$\begin{eqnarray}\displaystyle & & \displaystyle \sum ^{\flat }_{\substack{ |d|\leqslant Q \\ d\equiv 1~(\text{mod}~4)}}\biggl(\mathop{\sum }_{y<p<z}\frac{a(p)\unicode[STIX]{x1D712}_{d}(p)}{p}\biggr)^{2k}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{y^{2r}<n<z^{2r}}\frac{a(n)b_{2r}(n)}{n}\sum ^{\flat }_{\substack{ |d|\leqslant Q \\ d\equiv 1~(\text{mod}~4)}}\unicode[STIX]{x1D712}_{d}(n)\nonumber\\ \displaystyle & & \displaystyle \quad \ll Q\mathop{\sum }_{y^{r}<m<z^{r}}\frac{b_{2r}(m^{2})}{m^{2}}+Q^{1/2}\mathop{\sum }_{y^{2r}<n<z^{2r}}\frac{b_{2r}(n)}{n^{1/2}}\nonumber\\ \displaystyle & & \displaystyle \quad \ll Q\mathop{\sum }_{y^{r}<m<z^{r}}\frac{b_{2r}(m^{2})}{m^{2}}+Q^{1/2}\biggl(\mathop{\sum }_{y<p<z}\frac{1}{\sqrt{p}}\biggr)^{2r}.\end{eqnarray}$$
Furthermore, by (2.6) together with the fact that
$b_{r}(m)\leqslant r!$
, we obtain
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{y^{r}<m<z^{r}}\frac{b_{2r}(m^{2})}{m^{2}} & {\leqslant} & \displaystyle \binom{2r}{r}\mathop{\sum }_{y^{r}<m<z^{r}}\frac{b_{r}(m)^{2}}{m^{2}}\nonumber\\ \displaystyle & {\leqslant} & \displaystyle \frac{(2r)!}{r!}\mathop{\sum }_{y^{r}<m<z^{r}}\frac{b_{r}(m)}{m^{2}}=\frac{(2r)!}{r!}\biggl(\mathop{\sum }_{y<p<z}\frac{1}{p^{2}}\biggr)^{r}.\nonumber\end{eqnarray}$$
Inserting this bound in (3.4) and using that
$\sum _{y<p<z}1/\sqrt{p}\ll \sqrt{z}/\text{log}\,z$
completes the proof.☐
We now prove Theorem 1.5.
Proof of Theorem 1.5.
Let
$A\geqslant 1$
be fixed. First, it follows from Lemma 2.3, that, for any integer
$2\leqslant k\leqslant (\log Q)^{A}$
, there exist prime numbers
$q_{1},q_{2}$
such that
$Q^{1/3}<q_{1}<q_{2}<2Q^{1/3}$
, a character
$\unicode[STIX]{x1D713}$
of order
$k$
and conductor
$q_{1}q_{2}$
such that
$\unicode[STIX]{x1D713}(p)=1$
for all primes
$p\leqslant (\log Q)/(6A)$
.
Let
$\unicode[STIX]{x1D700}=\unicode[STIX]{x1D713}(-1)$
and let
$\ell <y\leqslant (\log Q)/(6A)$
be a real number to be chosen later. We consider the family of characters
$\{\unicode[STIX]{x1D713}\unicode[STIX]{x1D712}_{d}\}$
, where
$d$
ranges over the fundamental discriminants
$d\equiv 1~(\text{mod}~4)$
such that
$(d,P(y))=1$
and
$0<\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FF}d<Q^{1/3}$
. Since
$\ell ,d$
and
$q_{1}q_{2}$
are pairwise coprime,
$\unicode[STIX]{x1D713}\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}_{d}$
is primitive and
$\unicode[STIX]{x1D713}\unicode[STIX]{x1D712}_{d}$
is a primitive character of order
$k$
and conductor
$|d|q_{1}q_{2}\ll Q$
. Moreover, note that
$\unicode[STIX]{x1D713}\unicode[STIX]{x1D712}_{d}(-1)=\unicode[STIX]{x1D6FF}$
.
Let
$z=(\log Q)^{10}$
. Then it follows, from Proposition 2.4, that, for all but at most
$Q^{1/4}$
fundamental discriminants
$d$
with
$0<\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FF}d<Q^{1/3}$
,
$$\begin{eqnarray}L(1,\unicode[STIX]{x1D713}\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}_{d})=\mathop{\prod }_{p\leqslant z}\biggl(1-\frac{\unicode[STIX]{x1D713}(p)\unicode[STIX]{x1D709}(p)\unicode[STIX]{x1D712}_{d}(p)}{p}\biggr)^{-1}\biggl(1+O\biggl(\frac{1}{\log \log Q}\biggr)\biggr).\end{eqnarray}$$
Furthermore, by Proposition 3.1, there are at least
$\gg _{\ell }Q^{1/3}/(2^{\unicode[STIX]{x1D70B}(y)}\log y)$
fundamental discriminants
$d\equiv 1~(\text{mod}~4)$
such that
$0<\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FF}d<Q^{1/3}$
,
$(d,P(y))=1$
and
$\unicode[STIX]{x1D712}_{d}(p)=\unicode[STIX]{x1D709}(p)$
for all primes
$p\leqslant y$
such that
$p\nmid \ell$
.
Moreover, taking
$r=[\log Q/(60\log \log Q)]$
in Lemma 3.4, we obtain that the number of fundamental discriminants
$d$
with
$0<\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FF}d<Q^{1/3}$
such that
$$\begin{eqnarray}\biggl|\mathop{\sum }_{\log Q<p<z}\frac{\unicode[STIX]{x1D713}(p)\unicode[STIX]{x1D709}(p)\unicode[STIX]{x1D712}_{d}(p)}{p}\biggr|>\frac{1}{\log \log Q}\end{eqnarray}$$
is
$$\begin{eqnarray}\ll Q^{1/3}\biggl(\frac{4r\log \log Q}{\log Q}\biggr)^{r}\ll Q^{1/3}\exp \biggl(-\frac{\log Q}{30\log \log Q}\biggr),\end{eqnarray}$$
since
$\sum _{\log Q<p}1/p^{2}\leqslant 2/(\log Q\log \log Q)$
, by the prime number theorem.
Thus, choosing
$y=(\log Q)/(50A)$
, we deduce that there are
$$\begin{eqnarray}\gg _{\ell }Q^{1/3}\exp \biggl(-\frac{\log Q}{50\log \log Q}\biggr)\end{eqnarray}$$
fundamental discriminants
$d\equiv 1~(\text{mod}~4)$
such that
$0<\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FF}d<Q^{1/3}$
,
$(d,P(y))=1$
, the asymptotic formula (3.5) holds,
$\unicode[STIX]{x1D712}_{d}(p)=\unicode[STIX]{x1D709}(p)$
for all primes
$p\leqslant y$
with
$p\nmid \ell$
and
$$\begin{eqnarray}\biggl|\mathop{\sum }_{\log Q<p<z}\frac{\unicode[STIX]{x1D713}(p)\unicode[STIX]{x1D709}(p)\unicode[STIX]{x1D712}_{d}(p)}{p}\biggr|\leqslant \frac{1}{\log \log Q}.\end{eqnarray}$$
For these
$d$
, by (3.5),
$$\begin{eqnarray}\displaystyle L(1,\unicode[STIX]{x1D713}\unicode[STIX]{x1D709}\unicode[STIX]{x1D712}_{d}) & = & \displaystyle \mathop{\prod }_{\substack{ p\leqslant y \\ p\nmid \ell }}\biggl(1-\frac{1}{p}\biggr)^{-1}\exp \biggl(\mathop{\sum }_{y<p\leqslant \log Q}\frac{\unicode[STIX]{x1D713}(p)\unicode[STIX]{x1D709}(p)\unicode[STIX]{x1D712}_{d}(p)}{p}\biggr)\nonumber\\ \displaystyle & & \displaystyle \times \,\biggl(1+O\biggl(\frac{1}{\log \log Q}\biggr)\biggr)\nonumber\\ \displaystyle & = & \displaystyle \frac{\unicode[STIX]{x1D711}(\ell )}{\ell }\text{e}^{\unicode[STIX]{x1D6FE}}\log \log Q+O_{\ell ,A}(1),\nonumber\end{eqnarray}$$
as desired. ☐
