The sum of a hybrid arithmetic function over a sparse sequence

1.   Introduction

Let $H_{k}^{\ast}$ be the set of all normalized primitive holomorphic cusp form of even integral weight $k\geq2$ for the full modular group $SL_2(\mathbb{Z})$. The primitive holomorphic cusp form $f\in H_{k}^{\ast}$ at the cusp $z = \infty$ has the Fourier expansion

where $\lambda_{f}(n)$ is the $n$-th normalized Fourier coefficient. $\lambda_{f}(n)$ is real-valued and has the following multiplicative property

with $m, n\in \mathbb{N}^{+}$. In number theory, the study of the Fourier coefficient $\lambda_{f}(n)$ is of great significance and has attracted attention of many mathematicians. Let $d(n)$ be the Dirichlet divisor function. In 1974, Deligne ^[1] proved the Ramanujan-Petersson conjecture

In 1927, Hecke ^[2] established that

Subsequently, Hecke's result was refined by many scholars and the best result now is

where $\rho = -0.118\cdots$, proved by Wu ^[3]. In 1930, by their powerful method Rankin ^[4] and Selberg ^[5] proved

where $c$ is a positive constant depending on $f$. Recently, the exponent $\frac{3}{5}$ was improved to $\frac{3}{5}-\Delta$ with $\Delta\leq\frac{1}{560}$ by Huang ^[6]. There is also a long history on higher powers sums $\sum\limits_{n\leq x}\lambda_{f}^{l}(n)$ with $l\geq3$, and here we refer to the references ^[7,8] and the references therein for detailed historical descriptions.

Let $a, b, l\in \mathbb{Z}$. Let also $r_2(n)$ denote the number of representations of $n$ as $n = a^2 + b^2$, i.e.,

In 2013, Zhai ^[9] studied a related power sum over a sum of two squares and established the following asymptotic formula

where $P_2(t), P_4(t), P_6(t), P_8(t)$ are polynomials of degree $0, 1, 4, 13$, respectively,

for $l = 3, 5, 7$, and

Later, Xu ^[8] refined and generalized the results of Zhai ^[9]. Recently, Liu ^[10] further improved Zhai and Xu's results.

Let $\sigma(n)$ and $\phi(n)$ be the sum-of-divisors function and the Euler totient function, respectively. In 2015, Manski et al. ^[11] proved that

where $a', b', c' \in \mathbb{R}$, $2^{a'}\in \mathbb{N}$, $b'+c' > -r_{a'}$, $P_{m}(t)$ is a polynomial in $t$ of degree $m$ and $r_{a'}$ takes specific values as in [11, (2)].

Let $c, d \in \mathbb{R}$. Many scholars also studied the mean values of the arithmetic function $\lambda_{f}^{l}(n)\sigma^c(n)\phi^d(n)$ and we refer to ^[12] for historical results. In detail, Wei and Lao ^[12] proved that

where $P_2(t)$, $P_4(t)$, $P_6(t)$, $P_8(t)$ are polynomials in $t$ of degree 0, 1, 4, 13, respectively, $P_7(t)\equiv0$, and

In this paper, motivated by the above results we study the asymptotic behavior of the hybrid arithmetic function $\lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n)$ over the sparse sequence $\{n: n = a^2+b^2\}$. Define

where $1\leq l\leq 8$, $x$ is a sufficiently large real number, $a, b, l\in \mathbb{Z}$ and $c, d \in \mathbb{R}$. By combining some analytic methods with properties of some primitive automorphic $L$-functions we establish the following theorem.

Theorem 1.1. Let $f\in H_k^{\ast}$ and $\lambda_{f}(n)$ be the $n$-th normalized Fourier coefficient of $f$. Under the notations above, for any $\varepsilon > 0$, we have

where

for $l = 1, 3, 5, 7$, $A_2(t)$, $A_4(t)$, $A_6(t)$, $A_8(t)$ are polynomials in $t$ of degree 0, 1, 4, 13, respectively, and

For a random variable $X$ defined on a countable sample space $\mathbb{V}$, let $E(X)$ denote the mathematical expectation of $X$. With the help of Theorem 1.3, we can obtain the asymptotic mathematical expectation, denoted by $E\left(\lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n)r_2(n)\right)_{1\leq n\leq x}$, of $\lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n)$ over the sample space

Theorem 1.2. Under the same notations as in Theorem 1.3, we have

In the following Section 2, we give some preliminary lemmas. In Sections 3 and 4, we complete the proofs of Theorems 1.1 and 1.2, respectively.

Notation. Throughout this paper, we apply the letter $\varepsilon$ to represent a sufficiently small positive constant, whose value may change from statement to statement. The constants, both explicit and implicit, in Vinogradov symbols may depend on $\varepsilon$ and $f$.

2.   Preliminary lemmas

We first introduce some $L$-functions and then give some necessary lemmas. As usual, we define Riemann zeta function $\zeta(s)$ and Dirichlet $L$-function $L(s, \chi)$ as

for ${\rm Re}(s) > 1$, respectively. For the $n$-th normalized Fourier coefficient $\lambda_{f}(n)$, Deligne ^[1] showed that for any prime $p$ there are two complex numbers $\alpha_{f}(p)$ and $\beta_{f}(p)$ satisfying

Thus, the Hecke $L$-function associated to $f\in H_{k}^{\ast}$ can be represented as

Then, the $j$-th symmetric power $L$-function with $f\in H_{k}^{\ast}$ can be defined as, for ${\rm Re}(s) > 1$,

Note that

and

For ${\rm Re}(s) > 1$, the $j$-th symmetric power $L$-function twisted by the Dirichlet character $\chi$ is defined as

Recall (1.1). It was showed by Iwaniec ^[13] that

where $\chi(d)$ is the non-trivial Dirichlet character modulo 4. Let $r(n)$ denote $r_2(n)/4$. Since $\chi(n)$ is completely multiplicative, one has

Therefore, we can write

Now, we turn to give some necessary lemmas. From the recent deep results of Newton and Thorne ^[14,15], we know that all ${\rm sym}^j f$ with $j\in \mathbb{N}^+$ are automorphic cuspidal representations of $GL_{j+1}$. That is, the $j$-th symmetric power $L$-function with $L(s, {\rm sym}^j f)$ with $j\in \mathbb{N}^+$ has analytic continuation as an entire function in the whole plane and certain functional equations. Thus, $L(s, {\rm sym}^j f)$ with $j\in \mathbb{N}^+$ are general $L$-functions in sense of Perelli ^[16].

Lemma 2.1. For any

we have

Proof. The former three results can be found in the works [17, Theorem 5], [9, Lemma 2.3] and [18, Corollary 1.2], respectively. The last result follows from [16, Theorem 4] and [19, Proposition 1] plainly. □

Lemma 2.2. Let $\chi$ be the non-trivial Dirichlet character modulo 4. For any

one has

Proof. Since $\chi$ is the non-trivial Dirichlet character modulo 4, twisting by the character $\chi$ does not affect subconvexity bounds and convexity bounds of $L$-functions in the $t$'s aspect. □

Lemma 2.3. Let $f\in H_{k}^{\ast}$ and $\chi$ be the non-trivial Dirichlet character modulo 4. Then, for any $\varepsilon > 0$, $j\in \mathbb{N}^+$ and $|t|\geq 1$, we have

and

Proof. The first result is in [16, Lemma 13]. The second result follows from the first result by the same reason as in Lemma 2.2. □

Lemma 2.4. For any $U\geq U_0$, where $U_0$ is a sufficiently large constant, there exists $T^{\ast}\in(U, 2U),$ such that

where $C > 0$ is an absolute constant.

Proof. This result is proved by Ramachandra and Sankaranarayanan [20, Lemma 2]. □

Lemma 2.5. For any $\varepsilon > 0$, we have

uniformly for $T\geq1$.

Proof. This result was established by Ivić [21, Theorem 8.4 and (8.87)]. □

Lemma 2.6. Let

be a Dirichlet series with a finite abscissa of absolute convergence $\sigma_a$. Suppose there exists a real number $\alpha\geq0$ such that

(i)

where $\sigma_a < \sigma\leq \sigma_a+1,$ and that $B$ is a non-decreasing function satisfying

(ii)

where $n\geq 1.$ Then, for

we have

Proof. This is the well-known Perron's formula, which can be found in [22, Corollary 2.4]. □

Lemma 2.7. Let

Then, for $l = 1, \cdots, 8$, we have

where

where $\chi$ is the non-trivial Dirichlet character modulo 4 and $H_{l}(s)$ is absolutely convergent for

Proof. Here, we give the detailed proof for $l = 7$ as an example, since the remaining cases can be proven by following a similar argument.

For $l = 7$, due to the multiplicative property of $\lambda_{f}(n)$, $\sigma(n)$, $\phi(n)$ and $r(n)$, we have

Further, by the binomial theorem, (2.2) and (2.4) we have

where $H_7(s)$ converges absolutely and uniformly for ${\rm Re}(s) > c+d+\frac{1}{2}$. □

3.   Proof of Theorem 1.3

In this section, we shall give the proof of Theorem 1.1. Here, we shall give the detailed proofs for the cases $l = 7, 8$. For the cases $l = 1, 3, 5$, the proofs are similar to the proof of $l = 7$. For the cases $l = 2, 4, 6$, the proofs are similar to the proof of $l = 8$.

We first handle the case $l = 7$. Using Lemma 2.6 to $\sum\limits_{n\leq x}\lambda_{f}^7(n)\sigma^{c}(n)\phi^{d}(n)r(n)$, we get

Since, from Lemma 2.7, $G_7(s-c-d)H_7(s)\frac{x^{s}}{s}$ has no poles in the range

and $|t|\leq T$, by Cauchy's Residue Theorem we obtain

For the horizontal segments, since $H_7(s)$ is absolutely convergent in ${\rm Re}(s) > c+d+\frac{1}{2}$, by Lemmas 2.1 and 2.2, we have

Then, for the vertical segment, by Lemmas 2.1–2.3 and Cauchy's inequality, we have

Thus, according to (3.1) and (3.2), we get

Taking

we have

Then, we turn to the case $l = 8$. Using Lemma 2.6 to $\sum\limits_{n\leq x}\lambda_{f}^8(n)\sigma^{c}(n)\phi^{d}(n)r(n)$, we get

Since, from Lemma 2.7, $G_8(s-c-d)H_8(s)\frac{x^{s}}{s}$ only has one pole at $s = c+d+1$ of order $14$ in the range

and $|t|\leq T$, by Cauchy's Residue Theorem again we obtain

where $A_{13}'(t)$ is a polynomial in $t$ of degree 13.

For the horizontal segments, since $H_8(s)$ is absolutely convergent in

by Lemmas 2.1, 2.2 and 2.4, we have

Then, for the vertical segment, by Lemmas 2.1–2.3 and 2.5, we have

Thus, according to (3.3) and (3.4), we get

Taking

we have

4.   Proof of Theorem 1.2

For the Gauss circle problem, we have the following famous result

where $a, b\in \mathbb{Z}$. This result can be found in [13, Corollary 4.9]. Then, by the definition of mathematical expectation and Theorem 1.3, for $l = 1, 2, \cdots, 8$, one has

where the notations of $A_l(\log x)$ and $\theta_{l}$ are the same as ones in Theorem 1.3. Thus, we complete the proof of Theorem 1.2.

5.   Conclusions

In this paper, we establish the asymptotic formula of the sum of the hybrid arithmetic function $\lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n)$ over the sparse sequence $\{n: n = a^2+b^2\}$, i.e., $\sum_{n\leq x} \lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n)r_2(n)$ for $1\leq l\leq 8$, where $x$ is a sufficiently large real number, $\lambda_{f}(n)$ is the $n$-th normalized Fourier coefficient of the primitive holomorphic cusp form $f$ of even integral weight $k\geq2$ for $SL_2(\mathbb{Z})$, $\sigma(n)$ and $\phi(n)$ are the sum-of-divisors function and the Euler totient function, $r_2(n)$ denotes the number of representations of $n$ as $n = a^2 + b^2$, $a, b, l\in \mathbb{Z}$ and $c, d \in \mathbb{R}$. In addition, we also study the mathematical expectation of this hybrid arithmetic function. With the help of Theorems 1.3 and 1.2, we can understand the asymptotic behaviors of the hybrid arithmetic function $\lambda_{f}^{l}(n)\sigma^{c}(n)\phi^{d}(n)$ over the sparse sequence $\{n: n = a^2+b^2\}$ more precisely.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the referee for many useful comments on the manuscript. This work is supported by the National Natural Science Foundation of China (Grant No. 12171286).

Conflict of interest

The authors declare that they have no conflicts of interest.