1 Introduction and main results
Number-theoretic functions, also known as arithmetic functions, play a fundamental role in understanding the divisibility properties of integers, the distribution of prime numbers, and various other aspects of number theory. Several important arithmetic functions, such as the Möbius function
$\mu (n)$
, Euler’s totient function
$\varphi (n)$
, the divisor function
$d(n)$
, and the sums of divisor function, have deep applications in analytic number theory, algebra, quantum computing, and even string theory.
A particularly interesting class of arithmetic functions was introduced by Ivić, Tenenbaum, and Erdős in 1986–87; namely, arithmetic functions with a squarefull kernel (cf. [Reference Erdös and Ivić10, Reference Ivić and Tenenbaum20]). Any integer
$n \geq 1$
can be uniquely decomposed as
$n = q(n) t(n)$
, where
$\gcd (q(n), t(n)) = 1$
, the factor
$q(n)$
is squarefree, and
$t(n)$
is squarefull, meaning that
$p^2 \mid t(n)$
whenever
$p \mid t(n)$
. A nonnegative integer-valued arithmetic function
$\nu (n)$
is said to have a squarefull kernel, or be a
$2$
-full kernel function, if it satisfies
$\nu (n) = \nu (t(n))$
for all
$n \geq 1,$
and grows at most polynomially in any arbitrarily small exponent, i.e.,
$\nu (n) \ll n^{\epsilon }$
for any
$\epsilon> 0$
. The collection of such functions is denoted by
$K_2$
, as introduced by Ivić and Tenenbaum [Reference Ivić and Tenenbaum20].
This concept extends naturally to higher orders. Given an integer
$m \geq 2$
, any integer
$n \geq 1$
admits a unique factorization
$n = q(n) t(n)$
, where
$q(n)$
is m-free and
$t(n)$
is m-full, meaning that
$p^m \mid t(n)$
whenever
$p \mid t(n)$
. The class of nonnegative integer-valued functions satisfying
$\nu (n) = \nu (t(n))$
for all
$n \geq 1,$
and the growth condition
$ \nu (n) \ll n^{\epsilon }$
for any
$\epsilon> 0,$
is denoted by
$K_m$
. These functions, known as m-full kernel functions, generalize the framework developed for squarefull kernel functions. Note that
$K_m$
is not necessarily a class of multiplicative functions.
One of the most well-known arithmetic functions exhibiting completely random behaviour is given by the Fourier coefficients of a Hecke eigenform for the full modular group. In this article, we establish an asymptotic result concerning the power-saving error term of an arithmetic function involving the coefficients of an L-function associated with a Hecke eigenform, combined with a weighted m-full kernel function
$\nu (n) \in K_m$
. Recently, Venkatasubbareddy and Sankaranarayanan [Reference Venkatasubbareddy and Sankaranarayanan40, Reference Venkatasubbareddy and Sankaranarayanan41] established asymptotic formulae involving Hecke eigenvalues weighted by an m-full kernel function. To present our main results, we begin with an introduction to modular forms and the associated L-functions.
Let
$M_k(1)$
denote the space of modular forms of even weight
$k \geq 2$
for the full modular group
$SL_2(\mathbb {Z})$
, and let
$S_k(1)$
be the subspace of cusp forms in
$M_k(1)$
. A cusp form
$f \in S_k(1)$
is called a Hecke eigenform if it is a simultaneous eigenfunction for all Hecke operators.
Let
$\lambda _f(n)$
be the normalized nth Fourier coefficient in the Fourier expansion of
$f \in S_k(1)$
at infinity, given by
$$\begin{align*}f(z) = \sum_{n=1}^{\infty} \lambda_f(n)n^{\frac{k-1}{2}}e^{2 \pi i n z}, \end{align*}$$
for all
$z \in \mathbb {H}$
, where
$\mathbb {H}$
is the Poincaré upper half-plane. The function f is said to be normalized if
$\lambda _f(1) = 1$
. The set containing normalized
$f \in S_k(1)$
is denoted by
$S^{\ast }_k(1)$
. For
$f \in S^{\ast }_k(1)$
, the sequence
$\{ \lambda _f(n) \}_{n \ge 1}$
defines a real multiplicative function satisfying Deligne’s bound
$$\begin{align*}|\lambda_f(n)| \leq d(n) \ll n^{\epsilon}, \end{align*}$$
for any
$\epsilon> 0$
, where
$d(n)$
is the divisor function.
For a Hecke eigenform
$f \in S^{\ast }_k(1)$
, we define the associated L-function
$L(f,s)$
by
$$ \begin{align*} L(f,s) := \sum_{n=1}^{\infty} \frac{\lambda_f(n)}{n^s} = \prod_{p} \left(1 - \frac{\lambda_f(p)}{p^s} + \frac{1}{p^{2s}}\right)^{-1} = \prod_{p} \left(1 - \frac{\alpha_f(p)}{p^s}\right)^{-1} \left(1 - \frac{\beta_f(p)}{p^s}\right)^{-1}, \end{align*} $$
for
$\text {Re}(s)> 1$
, where for any prime p, there exist complex numbers
$\alpha _f(p)$
and
$\beta _f(p)$
such that
$$ \begin{align} \alpha_f(p) + \beta_f(p) = \lambda_f(p), \quad \text{and} \quad |\alpha_f(p)| = |\beta_f(p)| = 1, \quad \alpha_f(p) \beta_f(p) = 1. \end{align} $$
For more details, see Deligne [Reference Deligne9].
The average behaviour of arithmetic functions often exhibits inherent randomness, making them a subject of extensive study. Hecke [Reference Hecke15], in 1927, established the bound
$$\begin{align*}\sum_{n \leq X} \lambda_f(n) \ll X^{\frac{1}{2}}. \end{align*}$$
Subsequent works have refined this result, including
$$ \begin{align*} \sum_{n \leq X} \lambda_f(n) \ll_{f,\epsilon} \begin{cases} X^{\frac{11}{24}+\epsilon}, \quad \text{Kloosterman}\\ X^{\frac{4}{9}+\epsilon}, \quad \text{Davenport, Sali}\acute{\rm e}\\ X^{\frac{5}{12}+\epsilon}, \quad \text{Weil}\\ X^{\frac{1}{3}+\epsilon}, \quad \text{Deligne}\\ X^{\frac{1}{3}}, \quad \text{Hafner and Ivi}\acute{\rm c}\\ X^{\frac{1}{3}}(\log X)^{-0.0652+\epsilon}, \quad \text{Rankin}\\ X^{\frac{1}{3}}(\log X)^{-0.1185}, \quad \text{Wu.} \end{cases} \end{align*} $$
The second moment of Hecke eigenvalues was studied independently by Rankin [Reference Rankin34] and Selberg [Reference Selberg35] in 1930, who proved that
$$\begin{align*}\sum_{n \leq X} \lambda_f^2(n) = c_f X + O(X^{\frac{3}{5}}), \end{align*}$$
where
$c_f> 0$
is a constant depending on f. More recently, Huang [Reference Huang18] in 2021 improved this estimate by showing that
$$\begin{align*}\sum_{n \leq X} \lambda_f^2(n) = c_f X + O(X^{\frac{3}{5} - \delta}), \text{ where } \delta \leq \frac{1}{560}. \end{align*}$$
Beyond classical summation results, one can also investigate the behaviour of Hecke eigenvalues along polynomial sequences. For a monic quadratic polynomial
$q(x)$
, Blomer [Reference Blomer3] studied the first moment of
$\lambda _f(n)$
along values of
$ q(x)$
and established that
$$\begin{align*}\sum_{n \leq X} \lambda_f(q(n)) \ll X^{\frac{6}{7}+\epsilon}, \end{align*}$$
for any
$\epsilon> 0$
. Similar results have been obtained for polynomials in two variables of degree 2, such as
$q(x,y) = x^2 + y^2$
, in works by Banerjee and Pandey [Reference Banerjee and Pandey2], and Acharya [Reference Acharya1].
For
$\ell \geq 1$
, following the work of Garret and Harris [Reference Garrett and Harris11], we consider the multi-folded
$ L$
-function associated with the Hecke eigenform
$f \in S^{\ast }_k(1)$
, defined as
$$ \begin{align*} L(f \otimes \cdots \otimes_{\ell} f,s) & := \sum_{n = 1}^{\infty} \frac{\lambda_{f \otimes \cdots \otimes_{\ell} f}(n)}{n^{s}} \\ &= \prod_{p} \prod_{\sigma}\left(1-{\alpha_{f}^{\sigma(1)}(p)} {\alpha_{f}^{\sigma(2)}(p)} \cdots {\alpha_{f}^{\sigma(\ell)}(p)} {p^{-s}}\right)^{-1}, \end{align*} $$
where
$\sigma $
runs over the set of maps from
$\{1, 2, \ldots , \ell \}$
to
$\{1, 2\}$
satisfying
$$\begin{align*}\alpha^{\sigma(i)}_{f}(p) = \begin{cases} \alpha_f(p), & \text{if } \sigma(i) = 1, \\ \beta_f(p), & \text{if } \sigma(i) = 2. \end{cases} \end{align*}$$
It follows directly that the Fourier coefficient of the
$\ell $
-fold product L-function satisfies
$$ \begin{align} \lambda_{f \otimes f \otimes \cdots \otimes_{\ell} f}(p) = \lambda_{f}^{\ell}(p). \end{align} $$
For
$4 \leq \ell \leq 8$
, Venkatasubbareddy and Sankaranarayanan [Reference Venkatasubbareddy and Sankaranarayanan39] investigated the error term in the asymptotic formula for
$ \sum _{n \leq X} \lambda _{f \otimes f \otimes \cdots \otimes _{\ell } f}(n).$
Recent progress in this direction has been made in [Reference Hua16, Reference Huang18, Reference Lü and Sankaranarayanan28, Reference Tiwari and Godara37, Reference Venkatasubbareddy and Sankaranarayanan38], which serves as the primary motivation for this article.
Let
$i \ge 1$
be a fixed integer. For
$1 \leq j \leq i$
, let
$\alpha _j \geq 1$
and
$r_j \geq 1$
be positive integers. We define
$$ \begin{align} \Lambda^{i}_{\ell,f}(n) := \lambda^{\alpha_1}_{r_1,f \otimes \cdots \otimes_{\ell} f}(n) \cdots \lambda^{\alpha_i}_{r_i,f \otimes \cdots \otimes_{\ell} f}(n), \end{align} $$
where
$$ \begin{align*} \lambda_{r,f \otimes \cdots \otimes_{\ell} f}(n) =\sum_{n=n_1\cdots n_r} \lambda_{f \otimes \cdots \otimes_{\ell} f}(n_1) \cdots \lambda_{f \otimes \cdots \otimes_{\ell} f}(n_r). \end{align*} $$
Denote
$$\begin{align*}R_i := \prod_{j=1}^{i} r^{\alpha_j}_j \quad \text{ and } \quad A_i := \sum_{j=1}^{i} \alpha_j. \end{align*}$$
For
$n \in \mathbb {N}$
and
$r \le n$
,
${n \choose r }$
, denotes the binomial coefficient with the convention
${n \choose r } =0$
if
$r<0$
. Let
$[x]$
denote the greatest integer less than or equal to x. We now define the following terms:
$$ \begin{align} A_{i,\ell,1} &:= R_i \left({\ell A_i \choose \ell A_i/2} - {\ell A_i \choose \frac{\ell A_i}{2}-1}\right), \quad A_{i,\ell,2} := R_i \left({\ell A_i \choose \frac{\ell A_i}{2}-1} - {\ell A_i \choose \frac{\ell A_i}{2}-2}\right), \end{align} $$
$$ \begin{align} A_{i,\ell,3}(n) := R_i \left({\ell A_i \choose n} - {\ell A_i \choose n-1}\right). \end{align} $$
Similarly, we define
$$ \begin{align} B_{i,\ell,1} &:= R_i \left({\ell A_i \choose [\ell A_i/2]} - {\ell A_i \choose [\frac{\ell A_i}{2}]-1}\right). \end{align} $$
Let q be a positive integer. In this article, we establish an asymptotic formula for the following power sum:
$$ \begin{align} \sum_{\substack{n \leq X+1 \\ n\equiv 1(q)}} \Lambda^{i}_{\ell,f}(n), \end{align} $$
where
$\Lambda ^{i}_{\ell ,f}(n)$
is defined in (1.3). In particular, we prove the following result.
Theorem 1.1 Let
$f \in S_k(1)$
be a normalized Hecke eigenform of even weight
$k \geq 2$
. Let
$\ell , i \geq 1$
and
$q \geq 100$
be positive integers. For any
$\epsilon> 0$
and sufficiently large X, the following hold:
(i)
If
$\ell A_i$
is even and
$q \ll X^{\mathfrak {A}_{\ell }-\epsilon }$
, then we have
$$\begin{align*}\sum_{\substack{n \leq X+1 \\ n\equiv 1(q)}} \Lambda^{i}_{\ell,f}(n) = X P(\log X) \frac{\phi(q)^{A_{i,\ell,1}-1}}{q^{A_{i,\ell,1}}} + O\left(\frac{X^{1-\mathfrak{A}_{\ell}+\epsilon}q^{1+\epsilon}}{\phi(q)}\right), \end{align*}$$
uniformly, where
$P(t)$
is a polynomial of degree
$A_{i,\ell ,1}-1$
and
$$\begin{align*}\mathfrak{A}_{\ell} = \frac{138}{46A_{i,\ell,1} + 201A_{i,\ell,2} + 69 \sum_{n=0}^{[\ell A_i/2]-2} A_{i,\ell,3}(n) (\ell A_i - 2n + 1) + 46}. \end{align*}$$
Here,
$A_{i,\ell ,1}, A_{i,\ell ,2}$
, and
$A_{i,\ell ,3}(n)$
are given in (1.4) and (1.5).
(ii)
If
$\ell A_i$
is odd and
$q \ll X^{\frac {1}{2\mathfrak {B}_{\ell }}-\epsilon }$
, then we have
$$\begin{align*}\sum_{\substack{n \leq X+1 \\ n\equiv 1(q)}} \Lambda^{i}_{\ell,f}(n) \ll X^{1 - \frac{1}{2\mathfrak{B}_{\ell}} + \epsilon} q^{1+\epsilon}, \end{align*}$$
uniformly, where
$$\begin{align*}\mathfrak{B}_{\ell} = \frac{1}{2} B_{i,\ell,1} + \frac{1}{4} \sum_{n=0}^{[\ell A_i/2]-1} A_{i,\ell,3}(n) (\ell A_i - 2n + 1). \end{align*}$$
Here,
$B_{i,\ell ,1}$
and
$A_{i,\ell ,3}(n)$
are given in (1.6) and (1.5).
Next, we prove the following result on the shifted convolution of
$\Lambda ^{i}_{\ell ,f}(n)$
involving an m-full kernel function.
Theorem 1.2 Let
$f \in S_k(1)$
be a normalized Hecke eigenform of even weight
$k \geq 2$
. Let
$\ell , i \geq 1$
and
$m \geq 2$
be fixed positive integers. Let
$\nu (n) \in K_m$
be the m-full kernel function. For any
$\epsilon> 0$
and sufficiently large X, the following hold:
(i)
If
$\ell A_i$
is even, then we have
$$\begin{align*}\sum_{n \leq X} \nu(n) \Lambda^{i}_{\ell,f}(n+1) = X \widetilde{P}(\log X) + O\left(X^{1 - \left(\frac{\mathfrak{A}_{\ell}}{3} - \frac{\mathfrak{A}_{\ell}}{3m}\right) + \epsilon}\right), \end{align*}$$
where
$\widetilde {P}(t)$
is a polynomial of degree
$A_{i,\ell ,1}-1$
and
$\mathfrak {A}_{\ell }$
is given in Theorem 1.1
.
(ii)
If
$\ell A_i$
is odd, then we have
$$\begin{align*}\sum_{n \leq X} \nu(n) \Lambda^{i}_{\ell,f}(n+1) \ll X^{1 - \left(\frac{1}{6\mathfrak{B}_{\ell}} - \frac{1}{6m\mathfrak{B}_{\ell}}\right) + \epsilon}, \end{align*}$$
where
$\mathfrak {B}_{\ell }$
is given in Theorem 1.1
.
To establish Theorem 1.1, we first use the orthogonality relation to decompose the sum defined in (1.7) over principal, primitive, and non-primitive characters modulo q. As a second step, we prove a decomposition result concerning the Dirichlet series associated with the coefficients of the power sum in (1.7), expressing it in terms of automorphic L-functions. Subsequently, we determine the error term using subconvex bounds, integral moments, and Hölder’s inequality, along with Newton and Thorne’s celebrated result on the automorphy of the symmetric power lift of
$f \in S^{\ast }_k(1)$
. To prove Theorem 1.2, we use Theorem 1.1 along with several summation techniques.
Notations. Throughout the article, the symbol
$\epsilon $
represents an arbitrarily small positive quantity, which may vary from one occurrence to another but always remains
$\epsilon> 0$
. We use the notation
$s := \sigma + i t$
.
2 Preliminaries
This section provides a brief overview of twisted L-functions and symmetric power L-functions associated with Hecke eigenforms, emphasizing their fundamental properties. We also examine the relationships between the normalized Fourier coefficients of Hecke cusp forms and symmetric power L-functions. Furthermore, we discuss key results concerning subconvex bounds and integral moments. Finally, we present an essential result that plays a crucial role in establishing the main theorems of this article. From now onward,
$f \in S^{\ast }_k(1)$
.
The function
$\lambda _f(n)$
satisfies the following Hecke relation [Reference Iwaniec21, Equation (6.83)]:
$$ \begin{align} \lambda_f(m) \lambda_f(n) = \sum_{d \mid \gcd(m,n)} \lambda_f \left(\frac{mn}{d^2}\right), \end{align} $$
for all positive integers m and n.
For a Dirichlet character
$\chi $
modulo q, the twisted Hecke L-function is defined as
$$ \begin{align*} L(f \otimes \chi, s) := \sum_{n = 1}^{\infty} \frac{\lambda_{f}(n) \chi(n)}{n^{s}} \quad \text{ for } \operatorname{Re}(s)> 1. \end{align*} $$
It has the Euler product, satisfies a nice functional equation, and admits an analytic continuation to the entire complex plane [Reference Iwaniec21, Section 7.2].
For
$j \geq 2$
, the jth symmetric power L-function of degree
$j+1$
associated with f is given by
$$ \begin{align*} L({\text{sym}}^jf, s) = \prod_p \prod_{0 \leq r \leq j} \left(1 - \frac{\alpha^{j-r}_{f}(p) \beta^{r}_{f}(p)}{p^s}\right)^{-1} =: \sum_{n=1}^{\infty} \frac{\lambda_{{\text{sym}}^j f}(n)}{n^s}, \end{align*} $$
for
$\operatorname {Re}(s)> 1$
, where
$\alpha _{f}(p)$
,
$\beta _{f}(p)$
are complex numbers satisfying (1.1), and
$\lambda _{{\text {sym}}^j f}(n)$
is a real multiplicative function. For each prime p, we have
$$ \begin{align} \lambda_{{\text{sym}}^j f}(p) = \sum_{n=0}^{j} \alpha_{f}(p)^{j-n} \beta_{f}(p)^{n}. \end{align} $$
By Deligne’s bound, it is well-known that
$$ \begin{align*} |\lambda_{{\text{sym}}^j f}(n)| \leq d_{j+1}(n) \ll_{\epsilon} n^{\epsilon}, \end{align*} $$
for any
$\epsilon> 0$
, where
$d_{j+1}(n)$
denotes the
$(j+1)$
-fold divisor function.
Let
$\mathbb {A}_{\mathbb {Q}}$
be the ring of adeles of
$\mathbb {Q}$
. It is well known that
$f \in S^{\ast }_k(1)$
corresponds to an automorphic cuspidal representation
$\pi _f$
of
$\operatorname {GL}_2(\mathbb {A}_{\mathbb {Q}})$
, which in turn gives rise to an automorphic L-function
$L(\pi _f, s)$
that coincides with
$L(f, s)$
.
Langlands functoriality conjecture asserts that
$\pi _f$
gives rise to a symmetric power lift
$\text {sym}^j\pi _f$
, that is an automorphic representation whose L-function is the symmetric power L-function attached to f, i.e.,
$L(\text {sym}^j\pi _f,s)=L(\text {sym}^jf,s)$
. For the known cases, the lifts are cuspidal, namely, there exists an automorphic cuspidal self-dual representation, denoted by
$\text {sym}^j\pi _f$
of
$\operatorname {GL}_ {j+1}(\mathbb {A}_{\mathbb {Q}})$
, whose L-function is the same as
$L(\text {sym}^jf, s)$
. For
$1 \leq j \leq 8$
, the Langlands functoriality conjecture has been rigorously established through a series of works by Gelbart and Jacquet [Reference Gelbart and Jacquet12], Kim [Reference Kim24], Kim and Shahidi [Reference Kim and Shahidi25, Reference Kim and Shahidi26], Shahidi [Reference Shahidi36], and Clozel and Thorne [Reference Clozel and Thorne5–Reference Clozel and Thorne7]. More recently, for all
$j \ge 1$
, Newton and Thorne in [Reference Newton and Thorne31, Reference Newton and Thorne32] proved that
$\text {sym}^jf$
corresponds with a cuspidal automorphic representation of
$\operatorname {GL}_ {j+1}(\mathbb {A}_{\mathbb {Q}})$
. In particular, the authors established the existence of the symmetric power liftings
$\text {sym}^j\pi _f$
for all
$j \ge 1,$
regarding the automorphy of the symmetric power lifting for
$f \in S^{\ast }_k(1)$
. Furthermore, by combining the results of Newton and Thorne with those of Cogdell and Michel [Reference Cogdell and Michel8], it follows that the L-function
$L(\operatorname {sym}^j f, s)$
admits an analytic continuation as an entire function in the whole complex plane satisfying a functional equation of Riemann zeta-type. For further details, readers may refer to [Reference Iwaniec21, Chapter 5].
Similar to the twisted Hecke L-function, for each
$j \geq 2$
, we define the twisted jth symmetric power L-function as follows:
$$ \begin{align*} L(\text{sym}^{j} f \otimes \chi, s) = \sum_{n = 1}^{\infty} \frac{\lambda_{\text{sym}^{j} f}(n) \chi(n)}{n^{s}}. \end{align*} $$
This function shares similar properties with the jth symmetric power L-function. Let
$\zeta (s)$
and
$L(s, \chi )$
denote the Riemann zeta function and the Dirichlet L-function, respectively, i.e.,
$$ \begin{align*} \zeta(s) = \sum_{n = 1}^{\infty} n^{-s} \quad \text{ and } \quad L(s, \chi) = \sum_{n = 1}^{\infty} \chi(n) n^{-s} \quad \text{for } \text{Re}(s)> 1, \end{align*} $$
where
$\chi $
is a Dirichlet character modulo q.
We adopt the following conventions:
$$ \begin{align} \begin{aligned} L(\text{sym}^{0} f, s) &= \zeta(s), \quad & L(\text{sym}^{0} f \otimes \chi, s) &= L(s, \chi), \\ L(\text{sym}^{1} f, s) &= L(f, s), \quad & L(\text{sym}^{1} f \otimes \chi, s) &= L(f \otimes \chi, s). \end{aligned} \end{align} $$
The following two lemmas are crucial in handling the error terms in our results. For Lemma 2.1, we refer to [Reference Ivić19, Theorem 8.3] and [Reference Bourgain4], while Lemma 2.2 follows from [Reference Good13, Reference Jutila23]. Their proofs are omitted.
Lemma 2.1 For the twelfth integral moment of the Riemann zeta function, we have
$$ \begin{align} \int_{1}^{T} |\zeta(\tfrac{1}{2} +\epsilon+ it)|^{12} \text{d}t \ll T^{2 + \epsilon}, \end{align} $$
uniformly for
$T \ge 1$
. Furthermore, for
$\frac {1}{2} \leq \sigma \leq 2 + \epsilon $
, the Riemann zeta function satisfies the subconvex bound
$$ \begin{align} \zeta(\sigma + it) \ll (1+|t|)^{\max\{\frac{13}{42}(1-\sigma),0\} + \epsilon}. \end{align} $$
Lemma 2.2 For any
$\epsilon>0$
, the following bound holds for the L-function associated with
$f \in S^{\ast }_k(1)$
:
$$ \begin{align*} L(f, \sigma + it) \ll (1+|t|)^{\max\{\frac{2}{3}(1-\sigma), 0\} + \epsilon}, \end{align*} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 1$
and
$|t| \geq 1$
. Additionally, for the sixth integral moment, we have
$$ \begin{align*} \int_{1}^{T} |L(f, \tfrac{1}{2} +\epsilon+ it)|^{6}\text{d}t \ll T^{2+\epsilon}, \end{align*} $$
uniformly for
$T \geq 1$
.
We now state the following lemma, which will be used in the sequel. It follows from [Reference Heath-Brown14, Reference Perelli33, Reference Young42].
Lemma 2.3 Let
$\chi $
be a primitive character modulo q. If
$q \ll T^2$
, then we have the bound:
$$ \begin{align} L(\sigma + iT, \chi) \ll (q(1+|T|))^{\max\{\frac{1}{3}(1-\sigma), 0\} + \epsilon}, \end{align} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 2$
. Moreover, the fourth integral moment satisfies
$$ \begin{align} \int_{1}^{T} |L(\sigma + it, \chi)|^{4}\text{d}t \ll (qT)^{2(1-\sigma) + \epsilon}, \end{align} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 1+\epsilon $
and
$T \geq 1$
.
Lemma 2.4 Let
$\chi $
be a primitive character modulo q. If
$q \ll T^2$
, then we have the subconvex bounds:
$$ \begin{align} L(\text{sym}^2 f, \sigma + iT) \ll (1+|T|)^{\max\{\frac{6}{5}(1-\sigma), 0\} + \epsilon} \end{align} $$
and
$$ \begin{align} L(\text{sym}^2 f \otimes \chi, \sigma + iT) \ll (q(1+|T|))^{\max\{\frac{67}{46}(1-\sigma), 0\} + \epsilon}, \end{align} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 2$
and
$|T| \geq 1$
.
Proof The subconvex bounds in (2.8) and (2.9) follow from the Phragmén–Lindelöf convexity principle and results by Lin, Nunes, and Qi [Reference Lin, Nunes and Qi27], and Huang [Reference Huang18], respectively.
Let
$\mathfrak {L}(s, \mathcal {A})$
be a general L-function of degree
$2A$
in the sense of Perelli [Reference Perelli33].
Lemma 2.5 For the general L-function
$\mathfrak {L}(s, \mathcal {A})$
of degree
$2A$
, we have the subconvex bound
$$ \begin{align} \mathfrak{L}(\sigma+it, \mathcal{A}) \ll (\vert t\vert+1)^{\max\{A(1-\sigma),0\}+\epsilon}, \end{align} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 1+\epsilon $
and
$|t|\ge 1$
. Assume that the coefficients
$a_{n}$
of
$\mathfrak {L}(s, \mathcal {A})$
satisfy
$\sum _{n \le X}|a_n|^2 \ll X^{1+\epsilon }$
for any
$\epsilon>0$
. Then,
$$ \begin{align} \int_{T}^{2T} \vert \mathfrak{L}(\sigma+it, \mathcal{A}) \vert^2 \text{d}t \ll T^{2A(1-\sigma)+\epsilon}, \end{align} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 1+\epsilon $
and
$T \geq 1$
.
Proof The results (2.10) and (2.11) follow from the maximum modulus principle and [Reference Perelli33], respectively.
Lemma 2.6 Let
$\chi $
be a primitive character modulo q. For the general L-function
$\mathfrak {L}(s,\mathcal {A} \otimes \chi )$
of degree
$2A$
, we have
$$ \begin{align} \int_{T}^{2T} \vert \mathfrak{L}(\sigma+it,\mathcal{A} \otimes \chi) \vert^2 \text{d}t \ll T(qT)^{\omega(\sigma)+\epsilon} + (qT)^{2A(1-\sigma)+\epsilon}, \end{align} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 1+\epsilon $
and
$T \geq 1$
, where
$$\begin{align*}\omega(\sigma) = \begin{cases} 1, & \text{if } \sigma = 1/2, \\ 0, & \text{if } 1/2 < \sigma \leq 1. \end{cases} \end{align*}$$
Furthermore, we have
$$ \begin{align} \mathfrak{L}(\sigma+it,\mathcal{A} \otimes \chi) \ll (q|t|+1)^{\max\{A(1-\sigma),0\}+\epsilon}, \end{align} $$
uniformly for
$\frac {1}{2} \leq \sigma \leq 1+\epsilon $
and
$|t| \ge 1$
.
Proof The bound in (2.12) follows from Matsumoto [Reference Matsumoto29, Section 3] and Perelli [Reference Perelli33, Theorem 4]. The subconvex bound in (2.13) can be found in Jiang [Reference Jiang and Lü22, Lemma 2.4].
In the following lemma, we provide the decomposition of the Dirichlet series associated with the coefficients
$ \Lambda ^{i}_{\ell ,f}(n)$
twisted by
$\chi $
in terms of general L-functions.
Lemma 2.7 (Decomposition)
Define the Dirichlet series
$$ \begin{align*} R^i_{\ell}(s, \chi) := \sum_{n = 1}^{\infty} \frac{\Lambda^{i}_{\ell,f}(n) \chi(n)}{n^{s}}, \end{align*} $$
for
$\text {Re}(s) \gg 1$
. Then,
$$ \begin{align*} R^i_{\ell}(s, \chi) = L^i_{\ell}(s, \chi) U^i_{\ell}(s), \end{align*} $$
where
(i) If
$\ell A_i$
is even, then
$$ \begin{align*} L^{i}_{\ell}(s, \chi) = L(s, \chi)^{A_{i,\ell,1}} L(\text{sym}^{2} f \otimes \chi, s)^{A_{i,\ell,2}} \prod_{n=0}^{[\ell A_i/2]-2} L(\text{sym}^{\ell A_i-2n} f \otimes \chi, s)^{A_{i,\ell,3}(n)}, \end{align*} $$
where
$A_{i,\ell ,1}$
,
$A_{i,\ell ,2}$
, and
$A_{i,\ell ,3}(n)$
are given in (1.4) and (1.5). The function
$ U^{i}_{\ell }(s)$
converges absolutely and uniformly for
$\sigma> \frac {1}{2}$
with
$U^{i}_{\ell }(s) \neq 0$
when
$\sigma = 1$
.
(ii) If
$\ell A_i$
is odd, then
$$ \begin{align*} L^{i}_{\ell}(s, \chi) = L(f \otimes \chi, s)^{B_{i,\ell,1}} \prod_{n=0}^{[\ell A_i/2]-1} L(\text{sym}^{\ell A_i-2n} f \otimes \chi, s)^{A_{i,\ell,3}(n)}, \end{align*} $$
where
$B_{i,\ell ,1}$
and
$A_{i,\ell ,3}(n)$
are given in (1.6) and (1.5). The function
$U^{i}_{\ell }(s)$
converges absolutely and uniformly for
$\sigma> \frac {1}{2}$
with
$U^{i}_{\ell }(s) \neq 0$
when
$\sigma = 1$
.
Proof Recall that
$$\begin{align*}R_i := \prod_{j=1}^{i} r^{\alpha_j}_j \quad \text{ and } \quad A_i := \sum_{j=1}^{i} \alpha_j. \end{align*}$$
Using (1.3) and from [Reference Hua17, Eq. (24)], we have
$$ \begin{align*} \Lambda^{i}_{\ell,f}(p) &= \lambda^{\alpha_1}_{r_1,f \otimes \cdots \otimes_{\ell} f}(p)\cdots \lambda^{\alpha_i}_{r_i,f \otimes \cdots \otimes_{\ell} f}(p) = \left(\prod_{j=1}^{i}r^{\alpha_j}_j\right) \lambda_f(p)^{\big(\ell \sum_{j=1}^{i} \alpha_j\big)} \end{align*} $$
and
$$ \begin{align*} \lambda_f(p)^{\big(\ell \sum_{j=1}^{i} \alpha_j\big)} &= \sum_{n=0}^{[ \ell \sum_{j=1}^{i} \alpha_j / 2 ]} \left( {\ell \sum_{j=1}^{i} \alpha_j \choose n} - {\ell \sum_{j=1}^{i} \alpha_j \choose n-1} \right) \lambda_{\text{sym}^{(\ell \sum_{j=1}^{i} \alpha_j)-2n}f}(p) \\ &= \sum_{n=0}^{[ \ell A_i/2 ]} \left( {\ell A_i \choose n} - {\ell A_i \choose n-1} \right) \lambda_{\text{sym}^{\ell A_i-2n}f}(p). \end{align*} $$
This implies
$$ \begin{align*} \Lambda^{i}_{\ell,f}(p) &= R_i \sum_{n=0}^{[ \ell A_i/2 ]} \left( {\ell A_i \choose n} - {\ell A_i \choose n-1} \right) \lambda_{\text{sym}^{\ell A_i-2n}f}(p). \end{align*} $$
From [Reference Hua16],
$\lambda _{f \otimes \cdots \otimes _{\ell } f}(n)$
is multiplicative, so its r-fold Dirichlet convolution
$\lambda _{r,f \otimes \cdots \otimes _{\ell } f}(n)$
is also multiplicative. Hence, by definition,
$\Lambda ^{i}_{\ell ,f}(n)$
is multiplicative, and since
$\chi (n)$
is multiplicative,
$R^{i}_{\ell }(s, \chi )$
has an Euler product expansion:
$$ \begin{align*} R^{i}_{\ell}(s, \chi) &= \prod_{p} \left( 1+ \sum_{k \geq 1} \frac{\Lambda^{i}_{\ell,f}(p^k) \chi(p^k)}{p^{ks}} \right) \\ &= \prod_{n=0}^{[ \ell A_i/2 ]} L(\text{sym}^{\ell A_i-2n}f \otimes \chi, s)^{R_i \left( {\ell A_i \choose n} - {\ell A_i \choose n-1} \right)} U^{i}_{\ell}(s) \\ &=: L^{i}_{\ell}(s, \chi) U^{i}_{\ell}(s), \end{align*} $$
for
$\text {Re}(s)>1$
, where
$U^{i}_{\ell }(s)$
is absolutely and uniformly convergent for
$\sigma> \frac {1}{2}$
, and satisfies
$U^{i}_{\ell }(s) \neq 0$
when
$\sigma = 1$
. Then, using (2.3), we get the result.
3 Proof of Theorem 1.1
From Deligne’s bound (resp. Weil’s bound), we know that
$ \Lambda ^{i}_{\ell ,f}(n)\chi (n) \ll n^{\epsilon } $
for any
$\epsilon> 0$
. Let
$\chi $
be any Dirichlet character modulo q. By the orthogonality relation, we have
$$ \begin{align} \sum_{n\leq X+1\atop n\equiv 1(q)}\Lambda^{i}_{\ell,f}(n)=&\frac{1}{\phi(q)}\sum_{\chi(q)}\sum_{n\leq X+1}\Lambda^{i}_{\ell,f}(n)\chi(n) \notag\\ =&\frac{1}{\phi(q)}\sum_{n\leq X+1}\Lambda^{i}_{\ell,f}(n)\chi_{0}(n)+\frac{1}{\phi(q)}\sum_{n\leq X+1}\sum_{\chi\neq \chi_{0}\atop \chi \text{ is primitive}}\Lambda^{i}_{\ell,f}(n)\chi(n) \notag\\ &+\frac{1}{\phi(q)}\sum_{n\leq X+1}\sum_{\chi\neq \chi_{0}\atop \chi \text{ is non-primitive}}\Lambda^{i}_{\ell,f}(n)\chi(n) \notag\\ =&:\frac{1}{\phi(q)}\big(\textstyle{\sum_{1}+\sum_{2}+\sum_{3}}\big). \end{align} $$
For
$\sum _{1}$
, by Lemma 2.7 and applying Perron’s formula (see [Reference Murty30, p. 67]), we have
$$ \begin{align*} \sum_{n\leq X+1}\Lambda^{i}_{\ell,f}(n)\chi_{0}(n)=&\frac{1}{2\pi i}\int_{1+\epsilon-iT}^{1+\epsilon+iT}R^i_{\ell}(s, \chi_0)\frac{(X+1)^{s}}{s}\text{d}s+O\left(\frac{X^{1+\epsilon}}{T}\right). \end{align*} $$
Shifting the line of integration parallel to
$\text {Re}(s)=\frac {1}{2}+\epsilon =:\sigma _0$
and applying Cauchy’s residue theorem, we obtain
$$ \begin{align*} \sum_{n\leq X+1}\Lambda^{i}_{\ell,f}(n)\chi_{0}(n)&=\operatorname*{\mathrm{Res}}_{s=1}R^i_{\ell}(s, \chi_0)\frac{(X+1)^{s}}{s}\\ &\quad +\frac{1}{2\pi i}\left\{\int_{\sigma_0-iT}^{\sigma_0+iT}+\int_{\sigma_0+iT}^{1+\epsilon+iT}+\int_{1+\epsilon-iT}^{\sigma_0-iT}\right\}R^i_{\ell}(s, \chi_0)\frac{(X+1)^{s}}{s}\text{d}s \\ &\quad +O\left(\frac{X^{1+\epsilon}}{T}\right)\\ &=:\operatorname*{\mathrm{Res}}_{s=1}R^i_{\ell}(s, \chi_0)\frac{(X+1)^{s}}{s}+I_{1}+I_{2}+I_{3}+{O}\left(\frac{X^{1+\epsilon}}{T}\right). \end{align*} $$
It is easy to observe the following:
$$\begin{align*}L(s,\chi_{0})=\zeta(s)\prod_{p\mid q}\left(1-\frac{1}{p^{s}}\right) \end{align*}$$
and
$$\begin{align*}L(\text{{sym}}^{j}f\otimes\chi_{0},s)=L(\text{sym}^{j}f,s)\prod_{p\mid q}\prod_{0\leq m\leq j}\left(1-\frac{\alpha_{f}^{j-m}(p)\beta_{f}^{m}(p)}{p^{s}}\right). \end{align*}$$
for
$j \ge 1$
.
Let
$\chi $
be a non-primitive character modulo q, and let
$\chi ^{*}$
be a primitive character modulo
$q_1(\neq q)$
induced by
$\chi $
. Then, we have
$$\begin{align*}L(s,\chi)=L(s,\chi^{*})\prod_{p\mid q,\atop p\nmid q_{1}}\left(1-\frac{\chi^{*}(p)}{p^{s}}\right), \end{align*}$$
$$\begin{align*}L(\text{sym}^{j}f\otimes\chi,s)=L(\text{sym}^{j}f\otimes\chi^{*},s)\prod_{\substack{p\mid q,\\ p\nmid q_{1},\\ 0 \leq m \leq j}}\left(1-\frac{\alpha_{f}^{j-m}(p)\beta_{f}^{m}(p)\chi^{*}(p)}{p^{s}}\right) \text{ for } j \ge 1, \end{align*}$$
$$\begin{align*}\prod_{p\mid q,\atop p\nmid q_{1}}\left(1-\frac{\chi^{*}(p)}{p^{s}}\right)\ll q^{\epsilon} \quad \text{ for }~ \frac{1}{2}+\epsilon< \text{Re}(s)<1+\epsilon, \end{align*}$$
and
$$\begin{align*}\prod_{\substack{p\mid q,\\ p\nmid q_{1},\\ 0 \leq m \leq j}}\left(1-\frac{\alpha_{f}^{j-m}(p)\beta_{f}^{m}(p)\chi^{*}(p)}{p^{s}}\right)\ll q^{\epsilon} \quad \text{ for }~ \frac{1}{2}+\epsilon< \text{Re}(s)<1+\epsilon. \end{align*}$$
This implies that, for
$\sum _{2}$
and
$\sum _{3}$
, it is enough to consider the sum
$\sum _{2}$
. By Perron’s formula, we get
$$ \begin{align*} \sum_{\substack{n \leq X+1\\ \chi\neq \chi_{0}\\ \chi \text{ is primitive}}}\Lambda^{i}_{\ell,f}(n)\chi(n)=\frac{1}{2\pi i}\int_{1+\epsilon-iT}^{1+\epsilon+iT}R^i_{\ell}(s, \chi)\frac{(X+1)^{s}}{s}\text{d}s+O\left(\frac{X^{1+\epsilon}}{T}\right). \end{align*} $$
Moving the line of integration parallel to
$\text {Re}(s)=\frac {1}{2}+\epsilon =:\sigma _0$
, using Cauchy’s residue theorem and from Lemma 2.7, we have
$$ \begin{align*} \sum_{\substack{n \leq X+1\\ \chi\neq \chi_{0}\\ \chi \text{ is primitive}}}\Lambda^{i}_{\ell,f}(n)\chi(n)&=\frac{1}{2\pi i}\left\{\int_{\sigma_0-iT}^{\sigma_0+iT}+\int_{\sigma_0+iT}^{1+\epsilon+iT}+\int_{1+\epsilon-iT}^{\sigma_0-iT}\right\}R^i_{\ell}(s, \chi)\frac{(X+1)^{s}}{s}\text{d}s \\ &\quad +{O}\left(\frac{X^{1+\epsilon}}{T}\right)\\ &=:I_{4}+I_{5}+I_{6}+{O}\left(\frac{X^{1+\epsilon}}{T}\right). \end{align*} $$
Now, we estimate the integrals
$I_j$
(
$1 \leq j \leq 6$
) based on the parity of
$\ell A_i$
.
Case (i): Suppose
$\ell A_i$
is even. From Lemma 2.7 and using (2.4), (2.5), (2.8), and (2.11), and applying Hölder’s inequality, we get
$$ \begin{align*} |I_1| &\ll X^{\sigma_0}+ \int_{1}^{T} |L^i_{\ell}(\sigma_{0} +i t, \chi_0)| X^{\sigma_0}t^{-1} dt \nonumber\\ &\ll X^{\sigma_0}+{\frac{X^{\sigma_0}}{T}}\Bigg\{\underset{ 1 \le T_1 \le T }{\mathrm{sup}} \Bigg(\underset{{\frac{T_1}{2}} \le t \le T_1}{\mathrm{max}} \displaystyle \bigg|\zeta\bigg({\frac{1}{2}}+\epsilon+i t\bigg)^{A_{i,\ell,1}-6}\bigg|\bigg|{L\bigg(\text{sym}^{2}f,{\frac{1}{2}}+\epsilon+i t\bigg)}^{A_{i,\ell,2}}\bigg|\Bigg)\\ &\times \Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg| {\zeta\bigg({\frac{1}{2}}+\epsilon+i t\bigg)}\bigg|^{12} dt \Bigg)^{\frac{1}{2}} \\ &\times \Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg|\prod_{n=0}^{[\ell A_i/2]-2} {L\bigg( \text{sym}^{\ell A_i-2n}f,{\frac{1}{2}}+\epsilon+i t\bigg)}^{A_{i,\ell,3}(n)}\bigg|^2 dt \Bigg)^{\frac{1}{2}}\Bigg\}\\ &\ll X^{{\frac{1}{2}}+\epsilon}+\frac{X^{{\frac{1}{2}}+\epsilon}}{T}\Bigg\{T^{{\frac{13}{42}}\times {\frac{1}{2}}(A_{i,\ell,1}-6)+{\frac{6}{5}}\times {\frac{1}{2}}\times A_{i,\ell,2}+{\frac{2}{2}}+{\frac{1}{4}}\bigg(\sum_{n=0}^{[\ell A_i/2]-2}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\epsilon}\Bigg\}\\ &\ll X^{\frac{1}{2}+\epsilon}\displaystyle{T^{{\frac{13}{84}}A_{i,\ell,1}+{\frac{3}{5}}A_{i,\ell,2}+{\frac{1}{4}}\bigg(\sum_{n=0}^{[\ell A_i/2]-2}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+{\frac{1}{14}}-1+\epsilon}}. \end{align*} $$
One can notice that
$|I_2|+|I_3| \ll |I_1|$
. The function
$R^{i}_{\ell }(s,\chi _0)$
has a simple pole at
$s=1$
of order
$A_{i,\ell ,1}$
. Therefore, we have
$$ \begin{align*} \sum_{n\leq X+1}\Lambda^{i}_{\ell,f}(n)\chi_{0}(n) = X^{}P(\log X)\frac{\phi(q)^{A_{i,\ell,1}}}{q^{A_{i,\ell,1}}}+O\left( X^{\frac{1}{2}+\epsilon} T^{\gamma^{i}_{\ell}-1+\epsilon} \right)+O\bigg(\frac{X^{1+\epsilon}}{T}\bigg), \end{align*} $$
where
$P(\log X)$
is a polynomial of degree
$A_{i,\ell ,1}-1$
and
$$ \begin{align*}\gamma^{i}_{\ell} = \frac{13}{84}A_{i,\ell,1}+\frac{3}{5}A_{i,\ell,2}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-2}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\frac{1}{14}.\end{align*} $$
Now, we compute the vertical segment using (2.6), (2.7), (2.9), and (2.12), and applying Hölder’s inequality
$$ \begin{align*} & |I_4| \ll X^{\sigma_0}+ \int_{1}^{T} |L^i_{\ell}(\sigma_{0} +i t, \chi | X^{\sigma_0}t^{-1} dt \nonumber\\ & \ll X^{\sigma_0} \nonumber\\ &\quad +\frac{X^{\sigma_0}}{T} \Bigg\{\underset{1 \le T_1 \le T }{\mathrm{sup}} \Bigg(\underset{ \frac{T_1}{2} \le t \le T_1 }{\mathrm{max}} \displaystyle \bigg|L\bigg(\frac{1}{2}+\epsilon+i t,\chi\bigg)^{A_{i,\ell,1}-2}\bigg|\bigg|{L\bigg(\text{sym}^{2}f \otimes \chi,\frac{1}{2}+\epsilon+i t\bigg)}^{A_{i,\ell,2}}\bigg|\Bigg)\\ &\quad \times \Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg| {L\bigg(\frac{1}{2}+\epsilon+i t,\chi\bigg)}\bigg|^{4} dt \Bigg)^{\frac{1}{2}}\\ &\quad \times \Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg|\prod_{n=0}^{[\ell A_i/2]-2} {L\bigg( \text{sym}^{\ell A_i-2n}f \otimes \chi,\frac{1}{2}+\epsilon+i t\bigg)}^{A_{i,\ell,3}(n)}\bigg|^2 dt \Bigg)^{\frac{1}{2}}\Bigg\}\\ & \ll X^{\frac{1}{2}+\epsilon} +\frac{X^{\frac{1}{2}+\epsilon}}{T}\Bigg\{(qT)^{\frac{1}{3}\times\frac{1}{2}(A_{i,\ell,1}-2)+\frac{67}{46}\times \frac{1}{2}(A_{i,\ell,2})+\frac{1}{2}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-2}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\epsilon}\Bigg\}\\ &\ll X^{\frac{1}{2}+\epsilon}\displaystyle{(qT)^{\frac{1}{6}A_{i,\ell,1}+\frac{67}{92}A_{i,\ell,2}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-2}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\frac{1}{6}+\epsilon}}T^{-1}. \end{align*} $$
Also, one can check that
$|I_5|+|I_6|\ll |I_4|$
. Hence, from (3.1), we have
$$ \begin{align*} \sum_{n\leq X+1\atop n\equiv 1(q)}\Lambda^{i}_{\ell,f}(n)&= X^{}{P}_{}(\log X)\frac{\phi(q)^{A_{i,\ell,1}-1}}{q^{A_{i,\ell,1}}}+O\bigg(\frac{X}{\phi(q)T}\bigg)\\ & +O\bigg(X^{\frac{1}{2}+\epsilon}\displaystyle{(qT)^{\frac{1}{6}A_{i,\ell,1}+\frac{67}{92}A_{i,\ell,2}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-2}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\frac{1}{6}+\epsilon}}T^{-1}\bigg). \end{align*} $$
Take
$T=\frac {X^{\mathfrak {A}_{\ell }}}{q}$
to make the above error terms comparable. This yields
$$ \begin{align*} \sum_{n\leq X+1\atop n\equiv 1(q)}\Lambda^{i}_{\ell,f}(n)&= X^{}{P}_{}(\log X)\frac{\phi(q)^{A_{i,\ell,1}-1}}{q^{A_{i,\ell,1}}} +O\bigg(\frac{X^{1-\mathfrak{A}_{\ell}+\epsilon}q^{1+\epsilon}}{\phi(q)}\bigg) \end{align*} $$
for
$q \ll X^{\mathfrak {A}_{\ell }-\epsilon }$
, where
$$ \begin{align*}\mathfrak{A}_{\ell}=\frac{138}{46A_{i,\ell,1}+201A_{i,\ell,2}+69\big(\sum_{n=0}^{[\ell A_i/2]-2}A_{i,\ell,3}(n)(\ell A_i -2n+1)\big)+46}\end{align*} $$
and
${P}_{}(t)$
is a polynomial of degree
$A_{i,\ell ,1}-1$
.
Case (ii): Suppose
$\ell A_i$
is odd. Similar to Case (i), we get the following by making use of Lemma 2.2, (2.11), and applying Hölder’s inequality
$$ \begin{align*} |I_1| & \ll X^{\sigma_0}+ \int_{1}^{T} |L^{i}_{\ell}(\sigma_0+i t,\chi_0)|t^{-1}X^{\sigma_0} dt \nonumber\\ &\ll X^{\frac{1}{2}+\epsilon}+X^{\frac{1}{2}+\epsilon}\frac{1}{T} \Bigg\{\underset{1 \le T_1 \le T }{\mathrm{sup} } \Bigg(\underset{ \frac{T_1}{2} \le t \le T_1 }{\mathrm{max}} \displaystyle \bigg|L\bigg(f, \frac{1}{2}+\epsilon+i t\bigg)^{B_{i,\ell,1}-3}\bigg|\Bigg)\\ &\quad \times \Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg| {L\bigg(f, \frac{1}{2}+\epsilon+i t\bigg)}\bigg|^{6} dt \Bigg)^{\frac{1}{2}} \nonumber\\ &\quad \times\Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg|\prod_{n=0}^{[\ell A_i/2]-1} {L\bigg( \text{sym}^{\ell A_i-2n}f,\frac{1}{2}+\epsilon+i t\bigg)}^{A_{i,\ell,3}(n)}\bigg|^2 dt \Bigg)^{\frac{1}{2}}\Bigg\}\\ &\ll X^{\frac{1}{2}+\epsilon}+X^{\frac{1}{2}+\epsilon}\frac{1}{T}\Bigg\{T^{\frac{2}{3}\times \frac{1}{2}(B_{i,\ell,1}-3)+\epsilon}\times T^{{1}+\epsilon}\times \displaystyle T^{\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-1}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\epsilon}\Bigg\}\\ &\ll X^{\frac{1}{2}+\epsilon}{T^{\frac{1}{3}B_{i,\ell,1}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-1}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)-1+\epsilon}}. \end{align*} $$
One can check that
$|I_2|+|I_3|\ll |I_1|$
. Since the function
$R^{i}_{\ell }(s,\chi _0)$
is analytic in the obtained region. Therefore, we have
$$ \begin{align*} \sum_{n\leq X+1}\Lambda^{i}_{\ell,f}(n)\chi_{0}(n) &\ll X^{\frac{1}{2}+\epsilon}{T^{\frac{1}{3}B_{i,\ell,1}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-1}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)-1+\epsilon}} \\ &\quad +\frac{X^{1+\epsilon}}{T}, \end{align*} $$
Using Lemma 2.6 and applying Cauchy–Schwarz’s inequality, we have
$$ \begin{align*} |I_4| & \ll X^{\sigma_0}+ \int_{1}^{T} |L^{i}_{\ell}(\sigma_0+i t,\chi)|t^{-1}X^{\sigma_0} dt \nonumber\\ &\ll X^{\frac{1}{2}+\epsilon}+X^{\frac{1}{2}+\epsilon}\frac{1}{T} \Bigg\{\underset{1 \le T_1 \le T }{\mathrm{sup} } \Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg| {L\bigg(f\otimes \chi, \frac{1}{2}+\epsilon+i t\bigg)^{B_{i,\ell,1}}}\bigg|^{2} dt \Bigg)^{\frac{1}{2}}\\ &\times \Bigg(\int_{\frac{T_1}{2}}^{T_1}\bigg|\prod_{n=0}^{[\ell A_i/2]-1} {L\bigg( \text{sym}^{\ell A_i-2n}f\otimes \chi,\frac{1}{2}+\epsilon+i t\bigg)}^{A_{i,\ell,3}(n)}\bigg|^2 dt \Bigg)^{\frac{1}{2}}\Bigg\}\\ &\ll X^{\frac{1}{2}+\epsilon}+X^{\frac{1}{2}+\epsilon}\frac{1}{T}\Bigg\{(qT)^{\frac{1}{2}B_{i,\ell,1}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-1}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\epsilon}\Bigg\}\\ &\ll X^{\frac{1}{2}+\epsilon}{(qT)^{\frac{1}{2}B_{i,\ell,1}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-1}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\epsilon}}T^{-1}. \end{align*} $$
One can check that
$|I_5|+|I_6|\ll |I_4|$
. Therefore,
$$ \begin{align*} \sum_{\substack{n \leq X+1\\ \chi\neq \chi_{0}\\ \chi \text{ is primitive}}}\Lambda^{i}_{\ell,f}(n)\chi(n) &\ll X^{\frac{1}{2}+\epsilon}{(qT)^{\frac{1}{2}B_{i,\ell,1}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-1}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg)+\epsilon}}T^{-1} \\ &\quad +\frac{X^{1+\epsilon}}{T}. \end{align*} $$
Now, take
$T=\frac {X^{\frac {1}{2\mathfrak {B}_{\ell }}}}{q}$
to optimize the error term. This gives
$$ \begin{align*} \sum_{n\leq X+1\atop n\equiv 1(q)}\Lambda^{i}_{\ell,f}(n) \ll {X^{1-\frac{1}{2\mathfrak{B}_{\ell}}+\epsilon}q^{1+\epsilon}} \end{align*} $$
for
$q \ll X^{\frac {1}{2\mathfrak {B}_{\ell }}-\epsilon }$
, where
$$ \begin{align*}\mathfrak{B}_{\ell}=\frac{1}{2}B_{i,\ell,1}+\frac{1}{4}\bigg(\sum_{n=0}^{[\ell A_i/2]-1}A_{i,\ell,3}(n)(\ell A_i -2n+1)\bigg).\end{align*} $$
4 Proof of Theorem 1.2
Recall that
$$\begin{align*}\nu(n) = \nu(t(n)) \ll n^{\epsilon} \quad \text{(by definition)} \quad \text{and} \quad \Lambda^{i}_{\ell,f}(n) \ll n^{\epsilon}. \end{align*}$$
Let
$1 \leq H \leq X^{\frac {\mathfrak {A}_{\ell }}{3}}$
be a parameter to be chosen appropriately later. Then, we have
$$ \begin{align} &\sum_{n\leq X}\nu(n)\Lambda^{i}_{\ell,f}(n+1) \nonumber\\ &\quad = \sum_{n\leq X \atop t(n)\leq H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) + \sum_{n\leq X \atop t(n)> H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) \notag \\ &\quad = \sum_{n\leq X \atop t(n)\leq H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) + O\left(\sum_{H<t(n)\leq X} \nu(t(n)) \sum_{q(n)\leq \frac{X}{t(n)} \atop (q(n),t(n))=1} \Lambda^{i}_{\ell,f}(t(n)q(n)+1) \right) \notag \\ &\quad = \sum_{n\leq X \atop t(n)\leq H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) + O\left(\sum_{H<t(n)\leq X} t(n)^{\epsilon} \sum_{q(n)\leq \frac{X}{t(n)} \atop (q(n),t(n))=1} (t(n)q(n))^{\epsilon} \right) \notag \\ &\quad = \sum_{n\leq X \atop t(n)\leq H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) + O\left(\sum_{H<t(n)\leq X} t(n)^{\epsilon} \left( \frac{X}{t(n)} \right)^{1+\epsilon} \right) \notag \\ &\quad = \sum_{n\leq X \atop t(n)\leq H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) + O\left( X^{1+\epsilon} \sum_{H<t(n)\leq X} \frac{1}{t(n)} \right) \notag \\ &\quad = \sum_{n\leq X \atop t(n)\leq H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) + O(X^{1+\epsilon}H^{\frac{1}{m}-1}).\nonumber\\ \end{align} $$
Define
$g(l):=\sum \limits _{\alpha d^{m}=l}\mu (d)$
, so that
$g(q(n))=1$
. Then, we obtain
$$ \begin{align} & \sum_{n\leq X \atop t(n)\leq H}\nu(n)\Lambda^{i}_{\ell,f}(n+1) \nonumber \\ &\quad = \sum_{t(n)\leq H}\nu(t(n))\sum_{q(n)\leq \frac{X}{t(n)} \atop (q(n),t(n))=1} \Lambda^{i}_{\ell,f}(t(n)q(n)+1) \notag \\ &\quad = \sum_{t(n)\leq H}\nu(t(n))\sum_{q(n)\leq \frac{X}{t(n)} \atop (q(n),t(n))=1} g(q(n)) \Lambda^{i}_{\ell,f}(t(n)q(n)+1) \notag \\ &\quad = \sum_{t(n)\leq H}\nu(t(n))\sum_{q(n)\leq \frac{X}{t(n)} \atop (q(n),t(n))=1} \sum_{\alpha(n)d^{m}(n)=q(n)}\mu(d(n)) \Lambda^{i}_{\ell,f}(t(n)q(n)+1) \notag \\ &\quad = \sum_{t(n)\leq H}\nu(t(n))\sum_{d(n)\leq \left(\frac{X}{t(n)}\right)^{\frac{1}{m}} \atop (d(n),t(n))=1} \mu(d(n))\sum_{\alpha(n)\leq \frac{X}{t(n)d^{m}(n)} \atop (\alpha(n),t(n))=1} \Lambda^{i}_{\ell,f}(t(n)\alpha(n)d^{m}(n)+1) \notag \\ &\quad =: {\textstyle\sum_{1}^{*}} + {\textstyle\sum_{2}^{*}},\notag\\ \end{align} $$
where
$$ \begin{align*} {\textstyle\sum_{1}^{*}} &= \sum_{t(n)\leq H}\nu(t(n))\sum_{d(n)\leq H^{\frac{1}{m}} \atop (d(n),t(n))=1} \mu(d(n))\sum_{\alpha(n)\leq \frac{X}{t(n)d^{m}(n)} \atop (\alpha(n),t(n))=1} \Lambda^{i}_{\ell,f}(t(n)\alpha(n)d^{m}(n)+1), \end{align*} $$
and
$$ \begin{align*} {\textstyle\sum_{2}^{*}} &= \sum_{t(n)\leq H}\nu(t(n))\sum_{H^{\frac{1}{m}}<d(n)\leq \left(\frac{X}{t(n)}\right)^{\frac{1}{m}} \atop (d(n),t(n))=1} \mu(d(n))\sum_{\alpha(n)\leq \frac{X}{t(n)d^{m}(n)} \atop (\alpha(n),t(n))=1} \Lambda^{i}_{\ell,f}(t(n)\alpha(n)d^{m}(n)+1). \end{align*} $$
For
${\textstyle \sum _{2}^{*}}$
, we see that
$$ \begin{align} {\textstyle\sum_{2}^{*}} &\ll \sum_{t(n)\leq H} t(n)^{\epsilon} \sum_{d(n)\geq H^{\frac{1}{m}}} \sum_{\substack{\alpha(n)\leq \frac{X}{t(n)d^{m}(n)} \\ (\alpha(n),t(n))=1}} (t(n)\alpha(n)d^{m}(n))^{\epsilon} \notag \\ &\ll \sum_{t(n)\leq H} t(n)^{2\epsilon} \sum_{d(n)\geq H^{\frac{1}{m}}} d(n)^{m\epsilon} \left(\frac{X}{t(n)d^{m}(n)}\right)^{1+\epsilon} \notag \\ &\ll X^{1+\epsilon}H^{\frac{1}{m}-1}. \end{align} $$
Since
$\sum \limits _{\delta (n)\mid (\alpha (n),t(n))}\mu (\delta (n))=1$
, we can express
${\textstyle \sum _{1}^{*}}$
as
$$ \begin{align*} {\textstyle\sum_{1}^{*}} &= \sum_{t(n)\leq H} \nu(t(n)) \sum_{\substack{d(n)\leq H^{\frac{1}{m}} \\ (d(n),t(n))=1}} \mu(d(n)) \sum_{\delta(n)\mid t(n)} \mu(\delta(n)) \\ &\quad \times \sum_{\substack{\alpha_{1}(n) \delta(n) t(n) d^{m}(n) \leq X}} \Lambda^{i}_{\ell,f}(t(n)\alpha_{1}(n)\delta(n)d^{m}(n)+1). \end{align*} $$
Write
$$ \begin{align*} \sum_{\substack{\alpha_{1}(n) \delta(n) t(n) d^{m}(n) \leq X}} \Lambda^{i}_{\ell,f}(t(n)\alpha_{1}(n)\delta(n)d^{m}(n)+1) := \sum_{\substack{n\leq X+1 \\ n\equiv1(t(n)\delta(n)d^{m}(n))}} \Lambda^{i}_{\ell,f}(n). \end{align*} $$
From this point onward, we divide the proof into two cases based on the parity of
$\ell A_i$
.
Case (i): Suppose
$\ell A_i$
is even. By looking at Theorem 1.1(i), we split the sum as
$ \sum _{1}^{*} = \sum _{1}^{\prime } + \sum _{1}^{\prime \prime }, $
where
$$ \begin{align*} {\textstyle\sum_{1}^{\prime}} &= \sum_{t(n)\leq H} \nu(t(n)) \sum_{\substack{d(n)\leq H^{\frac{1}{m}} \\ (d(n),t(n))=1}} \mu(d(n)) \sum_{\delta(n)\mid t(n)} \mu(\delta(n)) \\ &\quad \times X {P}_{}(\log X) \frac{\phi(t(n)\delta(n)d^m(n))^{{A}_{i,\ell,1}-1}}{(t(n)\delta(n)d^{m}(n))^{{A}_{i,\ell,1}}}, \end{align*} $$
and
$$ \begin{align*} {\textstyle\sum_{1}^{\prime\prime}} &= O\Bigg(\sum_{t(n)\leq H} \nu(t(n)) \sum_{\substack{d(n)\leq H^{\frac{1}{m}} \\ (d(n),t(n))=1}} \mu(d(n)) \sum_{\delta(n)\mid t(n)} \mu(\delta(n)) \\ &\quad \times X^{1-\mathfrak{A}_{\ell}+\epsilon}\frac{(t(n)\delta(n)d^{m}(n))^{1+\epsilon}}{\phi(t(n)\delta(n)d^{m}(n))} \Bigg). \end{align*} $$
Note that
$\frac {q}{\phi (q)}\ll \log (\log q)$
. So, we see that
$$ \begin{align} {\textstyle\sum_{1}^{\prime\prime}} &\ll X^{1-\mathfrak{A}_{\ell}+\epsilon} H^{\frac{2}{m}} \end{align} $$
and
$$ \begin{align} {\textstyle\sum_{1}^{\prime}} &= X\widetilde{P}_{}(\log X) + O\left(X^{1+\epsilon}H^{\frac{1}{m}-1}\right). \end{align} $$
Combining (4.1)–(4.5), it follows that
$$ \begin{align*} \sum_{n\leq X} \nu(n) \Lambda^{i}_{\ell,f}(n+1) = X\widetilde{P}_{}(\log X) + O\left(X^{1+\epsilon}H^{\frac{1}{m}-1}\right) + O\bigg(X^{1-\mathfrak{A}_{\ell}+\epsilon}H^{\frac{2}{m}}\bigg). \end{align*} $$
By Theorem 1.1, we have
$q=\delta (n)t(n)d^{m}(n)$
with
$\delta (n), t(n), d^{m}(n) \leq H$
, which implies
$q \leq H^{3}$
. Moreover, by Theorem 1.1, q satisfies
$$\begin{align*}q \ll X^{\mathfrak{A}_{\ell}-\epsilon}. \end{align*}$$
We choose the optimal value of
$H = X^{\frac {\mathfrak {A}_{\ell }}{3}-\epsilon }$
to obtain
$$ \begin{align*} \sum_{n\leq X} \nu(n) \Lambda^{i}_{\ell,f}(n+1) = X\widetilde{P}_{}(\log X) + O\left(X^{1-(\frac{\mathfrak{A}_{\ell}}{3}-\frac{\mathfrak{A}_{\ell}}{3m})+\epsilon}\right), \text{ as desired.} \end{align*} $$
Case (ii): Suppose
$\ell A_i$
is odd. We proceed with similar arguments as in the previous case, with slight modifications. As a result, we have
$$ \begin{align*} \sum_{n\leq X} \nu(n) \Lambda^{i}_{\ell,f}(n+1) = O\left(X^{1 - \left(\frac{1}{6\mathfrak{B}_{\ell}} - \frac{1}{6m\mathfrak{B}_{\ell}}\right) + \epsilon} \right), \end{align*} $$
as desired.
Acknowledgments
We thank the anonymous referees for their careful review, detailed corrections, and valuable comments, which have significantly improved the clarity and readability of the manuscript. The second author gratefully acknowledges support from the Indian Institute of Technology Madras through the NFSG grant IP24251024MANFSC009034.
Data availability statement
This manuscript has no associated data.
Competing interests
The authors declare no competing interests.
