Approximate functional equation for the derivatives of functions in Selberg class

TL;DR

This paper extends the approximate functional equation, originally established for functions in the Selberg class, to include their derivatives, broadening the analytical tools available for such functions.

Contribution

The paper generalizes the approximate functional equation to derivatives of functions in the Selberg class, enhancing understanding of their analytical properties.

Findings

Derived the approximate functional equation for derivatives of Selberg class functions

Extended the applicability of the functional equation to higher derivatives

Provided a new analytical framework for studying Selberg class functions

Abstract

Let $F(s)$ be a function belonging to Selberg class. Chandrasekharan and Narasiman proved the approximate functional equation for $F(s)$. In this paper, we shall generalize this formula for the derivatives of $F(s)$.

Equations

$$\sum _{n\le x}{a}_F(n)=\{\begin{array}{ll}x{\sum }_{r=0}^{p_F-1}{c}_r(\log x{)}^r+O(x^{\frac{d_F-1}{d_F+1}+\epsilon }),& p_F\in {\mathbb{Z}}_{\ge 1},\\ O(x^{\frac{d_F-1}{d_F+1}+\epsilon }),& p_F\in {\mathbb{Z}}_{\le 0},\end{array}$$

$$F^{(m)}(s)=\sum _{r=0}^m(-1{)}^r(\genfrac{}{}{0ex}{}{n}{r}){\chi }_F^{(m-r)}(s){\overline{F}}^{(r)}(1-s)$$

$${\chi }_F(s)=\omega Q^{1-2s}\prod _{j=1}^q\frac{\Gamma ({\lambda }_j(1-s)+\overline{{\mu }_j})}{\Gamma ({\lambda }_{j}s+{\mu }_j)}.$$

$$F(s)=\sum _{n\le y}\frac{a_F(n)}{n^s}+{\chi }_F(s)\sum _{n\le y}\frac{a_F(n)}{n^{1-s}}+O(|t{|}^{\frac{d_F}{2}(1-\sigma )-\frac{1}{d_F}}(\log y{)}^{p_F})$$

$$L_f^{(m)}(s)=\sum _{n\le \frac{|t|}{2\pi }}\frac{{\lambda }_f(n)(-\log n{)}^m}{n^s}+{\chi }_{L_f}(s)\sum _{n\le \frac{|t|}{2\pi }}\frac{{\lambda }_f(n)(-\log n{)}^m}{n^{1-s}}+O(|t{|}^{1/2-\sigma +\epsilon }),$$

$${\int }_0^T|L_f^{(m)}(\sigma +it){|}^{2}dt$$

$$=\{\begin{array}{ll}{A}_{f}T(\log T{)}^{2m+1}+O(T(\log T{)}^{2m}),& \sigma =1/2,\\ T\sum _{n=1}^{\infty }\frac{|{\lambda }_f(n){|}^2(\log n{)}^{2m}}{n^{2\sigma }}+O(T^{2(1-\sigma )}(\log T{)}^{2m}),& \sigma \in (1/2,1),\\ T\sum _{n=1}^{\infty }\frac{|{\lambda }_f(n){|}^2(\log n{)}^{2m}}{n^{2\sigma }}+O((\log T{)}^{2m+2}),& \sigma =1.\end{array}$$

$$N_{L_f^{(m)}}(\sigma ,T)=O\left(\frac{T}{\sigma -1/2}\log \frac{1}{\sigma -1/2}\right)(T\to \infty )$$

$$N_{{\zeta }^{(m)}}(\sigma ,T)=O\left(\frac{T}{\sigma -1/2}\log \frac{1}{\sigma -1/2}\right)(T\to \infty )$$

$${\gamma }_j^{(r)}(s;\rho ):=$$

$$\times \prod _{j=1}^q\frac{\Gamma ({\lambda }_j(s+w)+{\mu }_j)}{\Gamma ({\lambda }_{j}s+{\mu }_j)}(\rho e^{-\frac{\pi }{2}i\mathrm{sgn}(t)}{)}^{\frac{d_F}{2}w}dw,$$

$${\delta }_j^{(r)}(s;\rho ):=$$

$$\times \prod _{j=1}^q\frac{\Gamma ({\lambda }_j(s+w)+\overline{{\mu }_j})}{\Gamma ({\lambda }_{j}s+\overline{{\mu }_j})}(\rho e^{-\frac{\pi }{2}i\mathrm{sgn}(t)}{)}^{\frac{d_F}{2}w}dw$$

$$g_F(s)=$$

$$A_j(s)$$

$$F^{(m)}(s)=$$

$$+\sum _{r=0}^m(-1{)}^r(\genfrac{}{}{0ex}{}{m}{r}){\chi }_f^{(m-r)}(s)\sum _{n=1}^{\infty }\frac{\overline{a_F(n)}(-\log n{)}^m}{n^{1-s}}{\phi }_0\left(\frac{n}{y_2}\right)+R_{\phi }(s)$$

$$R_{\phi }(s)$$

$$:=$$

$$+{\chi }_F(s)\sum _{r=0}^m(-1{)}^r(\genfrac{}{}{0ex}{}{m}{r})\sum _{\frac{y_2}{2}\le n\le 2y_2}\frac{\overline{a_F(n)}(-\log n{)}^r}{n^{1-s}}\times$$

$$\times \sum _{j=1}^l{\phi }^{(j)}\left(\frac{n}{y_2}\right){\left(-\frac{n}{y_2}\right)}^j{\delta }_j^{(r)}\left(1-s;\frac{1}{({\lambda }_1^{{\lambda }_1}\cdots {\lambda }_q^{{\lambda }_q}{)}^{\frac{2}{d_F}}|t|}\right)+$$

$$+O\left(y_1^{1-\sigma }(\log y_1{)}^{m+\max \{p_F-1,0\}}|t{|}^{-\frac{l}{2}}\Vert {\phi }^{(l+1)}{\Vert }_1\right)+$$

$$+O\left(y_2^{\sigma }|t{|}^{d_F(\frac{1}{2}-\sigma )-\frac{l}{2}}\Vert {\phi }_0^{(l+1)}{\Vert }_1\sum _{r=0}^m(\log y_2{)}^{r+\max \{p_F-1,0\}}(\log |t|{)}^{m-r}\right).$$

$$F^{(m)}(s)=$$

$$+O(y_1^{1-\sigma +\epsilon }|t{|}^{-\frac{1}{2}})+O(y_2^{\sigma +\epsilon }|t{|}^{d_F(\frac{1}{2}-\sigma )-\frac{1}{2}})$$

$$L_{f\times g}(s)=\sum _{n=1}^{\infty }\frac{{\lambda }_f(n)\overline{{\lambda }_g(n)}}{n^{-s}}.$$

$$L_{f\times g}^{(m)}(s)=$$

$$+\sum _{r=0}^m(-1{)}^r(\genfrac{}{}{0ex}{}{m}{r}){\chi }_{L_{f\times g}}^{(m-r)}(s)\sum _{n\le \frac{|t{|}^2}{4{\pi }^2}}\frac{\overline{{\lambda }_{f\times g}(n)}(-\log n{)}^r}{n^{1-s}}+O(|t{|}^{\frac{3}{2}-2\sigma +\epsilon })$$

$${\chi }_{L_{f\times g}}(s)=(2\pi {)}^{4s-2}\frac{\Gamma (1-s)\Gamma (1-s+k-1)}{\Gamma (s)\Gamma (s+k-1)}.$$

$$K_{\phi }(w):=$$

$$K_{\phi }(w)=K_{{\phi }_0}(-w).$$

$$K_{\phi }(w)=\frac{(-1{)}^{l+1}}{(w+1)\cdots (w+l)}{\int }_0^{\infty }{\phi }^{(l+1)}(\rho ){\rho }^{w+l}d\rho$$

$$\frac{F^{(r)}}{F}(s)=\sum _{1{\ell }_1+\cdots +r{\ell }_r=r}{C}_{{\ell }_1,\dots ,{\ell }_r}(G^{(1)}(s){)}^{{\ell }_1}\cdots (G^{(r)}(s){)}^{{\ell }_r}.$$

$$\sum _{n=1}^{\infty }\frac{1}{(s+n{)}^l}=l{\int }_1^{\infty }\frac{u-\{u\}}{(s+u{)}^{l+1}}du\ll {\int }_1^{\infty }\left(\frac{1}{|s+u{|}^l}+\frac{|s|}{|s+u{|}^{l+1}}\right)du.$$

$$|u+s|\ge \{\begin{array}{ll}|\mathrm{Im}(u+s)|\ge |s\sin \epsilon |,& u\le |s|\mathrm{and}\arg s\in [-\pi +\epsilon ,\pi -\epsilon ],\\ |\mathrm{Re}(u+s)|\ge |s|\cos \epsilon ,& u\le |s|\mathrm{and}\arg s\in [-\epsilon ,\epsilon ],\\ |\mathrm{Re}(u+s)|\ge u,& u\ge |s|,\end{array}$$

$$\ll {\int }_1^{|s|}\left(\frac{1}{|s\sin \epsilon {|}^l}+\frac{|s|}{|s\sin \epsilon {|}^{l+1}}\right)du+{\int }_{|s|}^{\infty }\left(\frac{1}{u^l}+\frac{|s|}{u^{l+1}}\right)du\ll \frac{1}{|s{|}^{l-1}}$$

Full text

1 Introduction

Selberg [8] introduced a class of zeta functions satisfying the following properties:

1. (a)

The function $F(s)$ is written as an absolutely convergent Dirichlet series $F(s)=\sum_{n=1}^{\infty}a_{F}(n)n^{-s}$ for ${\rm Re\;}s>1$. 2. (b)

There exists $m\in\mathbb{Z}_{\geq 0}$ such that $(s-1)^{m}F(s)$ is an entire function of finite order. 3. (c)

The function $\Phi(s)=Q^{s}\prod_{j=1}^{q}\Gamma(\lambda_{j}s+\mu_{j})F(s)$ satisfies $\Phi(s)=\omega\overline{\Phi}(1-s)$ where $q\in\mathbb{Z}_{\geq 1}$, $Q\in\mathbb{R}_{>0}$, $\lambda_{j}\in\mathbb{R}_{>0}$, $\mu_{j}\in\mathbb{C}:{\rm Re\;}\mu_{j}\geq 0$, $\omega\in\mathbb{C}:|\omega|=1$, and $\overline{X}$ denotes $\overline{X}(s)=\overline{X(\overline{s})}$. 4. (d)

The Dirichlet coefficients $a_{F}(n)$ satisfy $a_{F}(n)=O(n^{\varepsilon})$ for any $\varepsilon\in\mathbb{R}_{>0}$. 5. (e)

The function $\log F(s)$ is written as $\log{F(s)}=\sum_{n=1}^{\infty}{b_{F}(n)n^{-s}}$ where $b_{F}(n)$ satisfy $b_{F}(n)=0$ for $n\neq p^{r}\;(r\in\mathbb{Z}_{\geq 1})$ and $b_{F}(n)=O(n^{\theta})$ for some $\theta\in\mathbb{R}_{<1/2}$,

which is called the Selberg class $\mathcal{S}$. For example, we see that the Riemann zeta function $\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ and the $L$-function attatched to cusp forms $L_{f}(s)=\sum_{n=1}^{\infty}\lambda_{f}(n)n^{-s}$ belong to $\mathcal{S}$, where $f$ is a cusp form of weight $k$ given by $f(z)=\sum_{n=1}^{\infty}\lambda_{f}(n)n^{\frac{k-1}{2}}e^{2\pi inz}$. By Landau’s [5] result (see (15) of p.214), the conditions (a)–(d) of $\mathcal{S}$ imply that the average of $a_{F}(s)$ is approximated as

[TABLE]

where $c_{r}\in\mathbb{C}$, $p_{F}$ is the order of pole at $s=1$ for $F(s)$, and $d_{F}$ is given by $d_{F}=2\sum_{j=1}^{q}\lambda_{j}$ called the degree of $F$. The conditon (c) implies that the $m$-th derivatives of $F(s)$ holds

[TABLE]

for $s\in\mathbb{C}$ where $\chi_{F}(s)$ is given by

[TABLE]

Chandrasekharan and Narasiman [2] proved the approximate functional equation for a class of zeta functions (see Theorem 2 of p.53). In the Selberg class, this equation is written as

[TABLE]

under the condition $a_{n}\geq 0$ where $F\in\mathcal{S}$, $s=\sigma+it:0\leq\sigma\leq 1,|t|\geq 1$ and $y=(Q\lambda_{1}^{\lambda_{1}}\cdots\lambda_{q}^{\lambda_{q}})|t|^{{d_{F}}/{2}}$. On the other hand, the author [9] showed the approximate functional equation for the derivatives of $L$-function attached for cusp form as

[TABLE]

and introduced the mean value formula for $L_{f}^{(m)}(s)$ as

[TABLE]

where $\chi_{L_{f}}(s)=(-1)^{k/2}(2\pi)^{1-2s}\Gamma(1-s+\tfrac{k-1}{2})/\Gamma(s+\tfrac{k-1}{2})$ and $A_{f}$ is a positive constant depending on $f$. These results are generalized for those results for $L_{f}(s)$ proved by Good [4]. For $F\in\mathcal{S},\sigma>1/2$ and $T>0$, let $N_{F^{(m)}}(\sigma,T)$ be a number of zeros for $F^{(m)}(s)$ in the region ${\rm Re\;}s\geq\sigma$, $0<{\rm Im\;}s\leq T$. As the application of those results, the author [10] obtained a zero-density estimate for $L_{f}^{(m)}(s)$, which is

[TABLE]

for $\sigma>1/2$. This result corresponds the estimate of zero-density estimate for $\zeta^{(m)}(s)$

[TABLE]

for $\sigma>1/2$ which is shown by Aoki–Minamide [1]. However, zero-density estimates for derivatives for zeta-function belonging to Selberg class is not known.

In this paper we study the approximate functional equation for $F^{(m)}(s)$ in order to establish tools for estimating $N_{F^{(m)}}(\sigma,T)$ in next paper. To attain the above object we shall use Good’s [4] method and the result (1.1). Let $\mathcal{R}$ be a class of $C^{\infty}$-class functions $\varphi:[0,\infty)\to\mathbb{R}$ satisfying $\varphi(\rho)=1$ for $\rho\in[0,1/2]$ and $\varphi(\rho)=0$ for $\rho\in[2,\infty)$, which is called characteristic functions. Put $\varphi_{0}(\rho):=1-\varphi(1/\rho)$ and $\|\varphi^{(j)}\|_{1}=\int_{0}^{\infty}|\varphi^{(j)}(\rho)|d\rho$ where $\varphi^{(j)}$ is the $j$-th derivative function of $\varphi\in\mathcal{R}$. Then we see that $\varphi_{0}\in\mathcal{R}$ and $\|\varphi^{(j)}\|_{1}<\infty$. For $j\in\mathbb{Z}_{\geq 0}$, $r\in\{0,\dots,m\}$, $\rho\in\mathbb{R}_{>0}$ and $s\in\mathbb{C}:|t|\gg 1$, we define

[TABLE]

where ${\rm sgn}(t)=\pm 1$ if $t\gtrless 0$, $\mathcal{F}=\{(1/2\pm 1)-\sigma+\sqrt{|t|}e^{i\pi(\mp 1/2+\theta)}\mid\theta\in[0,1]\}\cup\{u-\sigma\pm i\sqrt{|t|}\mid u\in[-1/2,3/2]\}$ (double-sign corresponds), and $g_{F}(s)$ is given by

[TABLE]

Here $A_{j}(s)$ is given by

[TABLE]

where $\nu_{j}=0$ when ${\rm Re\;}\mu_{j}>\lambda_{j}/2$ and $\nu_{j}=[\lambda_{j}/2-{\rm Re\;}\mu_{j}]$ when ${\rm Re\;}\mu_{j}\leq\lambda_{j}/2$.

Then we obtain the approximate functional equation for $F^{(m)}(s)$ containing characteristic functions:

Theorem 1.1**.**

For any $F\in\mathcal{S}$, $\varphi\in\mathcal{R}$, $m\in\mathbb{Z}_{\geq 0}$, $l\in\mathbb{Z}_{>M_{F}}$, $s\in\mathbb{C}:\sigma\in[0,1],|t|\gg 1$ and $y_{1},y_{2}\in\mathbb{R}_{>0}:y_{1}y_{2}=(Q\lambda_{1}^{\lambda_{1}}\cdots\lambda_{q}^{\lambda_{q}})^{2}|t|^{d_{F}}$, we have

[TABLE]

where $M_{F}$ is some positive constant and $R_{\varphi}(s)$ is given by

[TABLE]

Next introducing new functions $\varphi_{\alpha}\in\mathcal{R}$ and $\xi\not\in\mathcal{R}$, replacing $\varphi$ to $\varphi_{\alpha}$ in the above theorem, and choosing $\alpha\in\mathbb{R}_{>0}$ in order to minimize the error term, we obtain the approximate functional equation for $F(s)$:

Theorem 1.2**.**

For any $F\in\mathcal{S}$, $m\in\mathbb{Z}_{\geq 0}$ and $s=\sigma+it:0\leq\sigma\leq 1,|t|\gg 1$ we have

[TABLE]

*where $y_{1},y_{2}\in\mathbb{R}_{>0}:y_{1}y_{2}=(Q\lambda_{1}^{\lambda_{1}}\cdots\lambda_{q}^{\lambda_{q}})^{2}|t|^{d_{F}}$. By choosing

$y_{1}=y_{2}=(Q\lambda_{1}^{\lambda_{1}}\cdots\lambda_{q}^{\lambda_{q}})|t|^{d_{F}/2}$, the error terms of (1.9) are $O(|t|^{\frac{d_{F}}{2}(1-\sigma)-\frac{1}{2}+\varepsilon})$.*

As an example of Theorem 1.2, we shall give the approximate functional equation for derivatives of Rankin-Selberg $L$-function. Let $f$ and $g$ be holomorphic cusp forms of weight $k$ with respect to $SL_{2}(\mathbb{Z})$ given by $f(z)=\sum_{n=1}^{\infty}\lambda_{f}(n)n^{\frac{k-1}{2}}e^{2\pi inz}$ and $g(z)=\sum_{n=1}^{\infty}\lambda_{g}(n)n^{\frac{k-1}{2}}e^{2\pi inz}$, the Rankin-Selberg $L$-function is defined by

[TABLE]

By results of Rankin [6], Selberg [7] and Delinge [3], we see that $L_{f\times g}(s)$ belongs to $\mathcal{S}$. Hence we obtain

[TABLE]

where $m\in\mathbb{Z}_{\geq 0}$, $s=\sigma+it:\sigma\in[0,1],|t|\gg 1$, and $\chi_{L_{f\times g}}(s)$ is given by

[TABLE]

In next section we shall show preliminary lemmas to prove Theorems 1.1 and 1.2. Using these lemmas we shall give Theorems 1.1 and 1.2 in Section 3 and 4 respectively.

2 Some Lemmas

First for $\varphi\in\mathcal{R}$ we define

[TABLE]

Then $K_{\varphi}(w)$ has the following properties:

Lemma 2.1** ([4, LEMMA 3]).**

The function $K_{\varphi}(w)$ is analytically continued to the whole $w$-plane. Furthermore $K_{\varphi}(w)$ has the functional equation

[TABLE]

and the integral representation

[TABLE]

for any $l\in\mathbb{Z}_{\geq 0}$. Especially $K_{\varphi}(0)=1$.

Next to approximate $(\chi_{F}^{(r)}/\chi_{F})(s)$, we shall use the following lemma:

Lemma 2.2** ([9, Lemma 2.3]).**

Let $F$ and $G$ be holomorphic functions in the region in $D$ satisfying $\log F(s)=G(s)$ and $F(s)\neq 0$ for $s\in D$. Then for any $r\in\mathbb{Z}_{\geq 1}$ there exist $\ell_{1},\dots,\ell_{r}\in\mathbb{Z}_{\geq 0}$ and $C_{\ell_{1},\dots,\ell_{r}}\in\mathbb{Z}_{\geq 0}$ such that

[TABLE]

for $s\in D$. Especially $C_{r,0\cdots,0}=1$.

Before approximating $(\chi_{F}^{(r)}/\chi_{F})(s)$ we shall show the following formulas:

Lemma 2.3**.**

Let $D=\{z\in\mathbb{C}\mid{\rm Re\>}z<\delta,|{\rm Im\>}z|<1\}$ where $\delta\in\mathbb{R}_{\geq 0}$. For any $s\in\mathbb{C}\setminus D$ we satisfy the following formulas:

1. (i)

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{(s+n)^{l}}=\begin{cases}O(|t|^{-(l-1)}),&|t|\gg 1,\\ O(1),&|t|\ll 1,\end{cases}$*  where $l\in\mathbb{Z}_{\geq 2}$.* 2. (ii)

$\displaystyle\sum_{n=1}^{\infty}\left(\frac{1}{s+n}-\frac{1}{n}\right)=\begin{cases}-\log|t|+i{\pi}{\rm sgn}(t)/2-\gamma+O(|t|^{-1}),&|t|\gg 1,\\ O(1),&|t|\ll 1,\end{cases}$

where $\gamma$ is the Euler’s constant given by $\gamma=1-\int_{1}^{\infty}\{u\}u^{-2}du$.

Proof.

Using partial summation we have

[TABLE]

The estimates

[TABLE]

give that the right-hand side of (2.4) is

[TABLE]

for any $s\in\mathbb{C}\setminus D$ and $l\in\mathbb{Z}_{\geq 2}$. Combining (2.4) and (2.6), and estimating $|s|^{-(l-1)}$ we obtain the formula (i). Similally to (i), we calculate

[TABLE]

Then for $s\in\mathbb{C}\setminus D$ the first and second term of right-hand side of (2.7) are

[TABLE]

and

[TABLE]

respectively, where (2.5) and $\log s=\log|t|+i\pi{\rm sgn}(t)/2+O(|t|^{-1})$ (see [4, p.335]) were used. By combining (2.7)–(2.9) the formula (ii) is obtained. ∎

Using the infinite product of $\Gamma(s)$ and applying the above lemma with $F=\chi_{F}$, we obtain the approximate formula for $(\chi_{F}^{(r)}/\chi_{F})(s)$ as follows:

Lemma 2.4**.**

For any $F\in\mathcal{S}$ and $r\in\mathbb{Z}_{\geq 0}$ the function $(\chi_{F}^{(r)}/\chi_{F})(s)$ has pole of order $r$ at $s=-(\mu_{j}+n)/\lambda_{j},\>1+(\overline{\mu_{j}}+n)/\lambda_{j}$ where $n\in\mathbb{Z}_{\geq 0}$ and $j\in\left\{1,\dots,q\right\}$. Put

[TABLE]

where $a,b\in\mathbb{R}$ and $\delta\in\mathbb{R}_{>0}$. Then for any $s\in\mathbb{C}\setminus(E_{1}\cup E_{2})$ we have

[TABLE]

where $C_{F}=(Q\lambda_{1}^{\lambda_{1}}\cdots\lambda_{q}^{\lambda_{q}})^{2}$.

Proof.

First we check the location and order of pole for $(\chi_{F}^{(m)}/\chi_{F})(s)$. Taking logarithmic differentiation in the both-hand side of $(1/\Gamma)(s)=se^{\gamma s}\prod_{n=1}^{\infty}(1+s/n)e^{-s/n}$ and (1.3) we have

[TABLE]

respectively. Put $G^{(l)}(s)=(d^{l-1}/ds^{l-1})G^{(1)}(s)$ and let $G^{(1)}(s)$ be the right-hand side of (2.11). Applying (2.10) to (2.11), we get

[TABLE]

Hence $G^{(l)}(s)$ has pole of order $l$ at $s=-(\mu_{j}+n)/\lambda_{j},\;1+(\overline{\mu_{j}}+n)/\lambda_{j}\;(n\in\mathbb{Z}_{\geq 0})$. By using Lemma 2.2 the first statement of Lemma 2.4 is showed.

Lastly we shall approximate $G^{(l)}(s)$. Since ${\rm Re}(\lambda_{j}s+\mu_{j})<\delta$, $|{\rm Im}(\lambda_{j}s+\mu_{j})|<1$ for $s\in E_{1}$ and ${\rm Re}(\lambda_{j}(1-s)+\mu_{j})<\delta$, $|{\rm Im}(\lambda_{j}(1-s)+\overline{\mu_{j}})|<1$ for $s\in E_{2}$, Lemma 2.3 gives that

[TABLE]

for $s\in\mathbb{C}\setminus(E_{1}\cup E_{2})$. Especially in the case of $j=1$ and $|t|\gg 1$, since ${\rm sgn}({\rm Im}(\lambda_{j}s+\mu_{j}))+{\rm sgn}({\rm Im}(\lambda_{j}(1-s)+\overline{\mu_{j}}))=0$, $G^{(1)}(s)$ is approximated as

[TABLE]

for $s\in\mathbb{C}\setminus(E_{1}\cup E_{2})$ and $|t|\gg 1$. By Lemma 2.2 a desired approximation is obtained:

[TABLE]

∎

Next in order to estimate the gamma-factors of (1.5) and (1.6), we shall use the following estimates:

Lemma 2.5** ([4, Lemma 2 of p.334]).**

For $a,b\in\mathbb{R}$ put $D:=\{z\in\mathbb{C}\mid a\leq{\rm Re\>}z\leq b\}$ and $E_{-}:=\{z\in\mathbb{C}\mid{\rm Re\>}z<1/2,|{\rm Im\>}z|<1\}$. Then for any fixed $s\in D$ and $C_{0}\in\mathbb{R}_{>0}$ we have

[TABLE]

where $C_{1}$ and $C_{2}$ are constants.

Replace $s\mapsto\lambda_{j}s+\mu_{j}$ and $w\mapsto\lambda_{j}w$ in the above lemma. Using the trivial estimate $(1+|\lambda_{j}(t+v)+{\rm Im\>}\mu_{j}|)^{\lambda_{j}(\sigma+u)+{\rm Re\>}\mu_{j}-1/2}\asymp\lambda_{j}^{\lambda_{j}u}(1+|t+v|)^{\lambda_{j}(\sigma+u)+{\rm Re\>}\mu_{j}-1/2}$ for $s\in D$ and $s+w\in D\setminus E_{1}$, and multiplying the above formula for $j\in\{1,\dots,q\}$, a desired estimate is obtained:

Lemma 2.6**.**

Let $D$ and $E_{1}$ be those of Lemma 2.4 respectively. Then for any fixed $s\in D$ and $c_{0}\in\mathbb{R}_{>0}$ we have

[TABLE]

where $e_{F}:=2\sum_{j=1}^{q}{\rm Re\;}\mu_{j}$ and $c_{1}$, $c_{2}$ are constants depending on $\lambda_{1},\mu_{1},\dots,\lambda_{q},\mu_{q}$.

By using Stirling’s formula and residue theorem the functions $\gamma_{j}^{(r)}(s;\rho)$ and

$\delta_{j}^{(r)}(s;\rho)$ are approximate as follows:

Lemma 2.7**.**

For any $j,r\in\mathbb{Z}_{\geq 0}$ and $s\in\mathbb{C}:|t|\gg 1$ we have

[TABLE]

The function ${\delta}_{j}^{(r)}(s;(\lambda_{1}^{\lambda_{1}}\cdots\lambda_{q}^{\lambda_{q}})^{-\frac{2}{d_{F}}}|t|^{-1})$ equals the right hand side of (2.13).

Proof.

Since $|w|\ll\sqrt{|t|}$ for $w\in\mathcal{F}$, from Lemma 2.6 we can obtain a desired formula in the case of $j\in\mathbb{Z}_{\geq 2}$:

[TABLE]

Using Cauchy’s residue theorem we have $\gamma_{0}^{(r)}(s,\rho)=(\chi_{F}^{(r)}/\chi_{F})(1-s)$ and

[TABLE]

where we used $|t|e^{\frac{\pi}{2}i{\rm sgn}(t)}=it$. It is clear that

[TABLE]

Stirling’s formula $\Gamma(s)=\sqrt{2\pi}s^{s-1/2}e^{-s}(1+O(|s|^{-1}))$ and the trivial approximation $\lambda_{j}s+\mu_{j}=i\lambda_{j}t(1+O(|t|^{-1}))$ give that

[TABLE]

Combining (2.14)–(2.16) and using Lemma 2.4 we obtain

[TABLE]

By the same discussion, the approximate formula of ${\delta}_{j}^{(r)}(s;1/((\lambda_{1}^{\lambda_{1}}\cdots\lambda_{q}^{\lambda_{q}})^{\frac{2}{d_{F}}}|t|))$ is also obtained. ∎

Finally, in order to prove Theorem 1.2 from Theorem 1.1, we introduce new functions. For $\varphi\in\mathcal{R}$, $\alpha\in\mathbb{R}_{\geq 0}$ and $|t|\gg 1$ we set

[TABLE]

Then these function have the following properties:

Lemma 2.8** ([4, (12)–(15)]).**

For any $\alpha\in\mathbb{R}_{\geq 0}$ and $\varphi\in\mathcal{R}$ we satisfy the following statements (i)–(iv):

1. (i)

$\varphi_{\alpha},\varphi_{0\alpha}\in\mathcal{R}$. 2. (ii)

$(\varphi_{\alpha}-\xi)(\rho)=0,\ (\varphi_{0\alpha}-\xi)(\rho)=0$* for $\rho\in[0,1-(2|t|^{\alpha})^{-1}]\cup[1+|t|^{-\alpha},\infty)$.* 3. (iii)

$\varphi_{\alpha}^{(j)}(\rho)=0,\>\>\varphi_{0\alpha}^{(j)}(\rho)=0$* for $j\in\mathbb{Z}_{\geq 1}$ and $\rho\in[0,1-(2|t|^{\alpha})^{-1}]\cup[1+|t|^{-\alpha},\infty)$.* 4. (iv)

$\varphi_{\alpha}^{(j)}(\rho)\ll|t|^{\alpha j},\ \varphi_{0\alpha}^{(j)}(\rho)\ll|t|^{\alpha j},\ \|\varphi_{\alpha}^{(j)}\|_{1}\ll|t|^{\alpha(j-1)},\ \|\varphi_{0\alpha}^{(j)}\|_{1}\ll|t|^{\alpha(j-1)}$* for $\rho\in[0,\infty)$ and $j\in\mathbb{Z}_{\geq 0}$.*

3 Proof of Theorem 1.1

First for $s=\sigma+it:\sigma\in[0,1],|t|\gg 1$, we shall use Cauchy’s integral theorem in the region

[TABLE]

Then from (1.2) and Lemma 2.2 the following lemma is obtained:

Proposition 3.1**.**

For any $m\in\mathbb{Z}_{\geq 0}$, $F\in\mathcal{S}$, $s=\sigma+it:\sigma\in[0,1],|t|\gg 1$, $\varphi\in\mathcal{R}$ and $x\in\mathbb{R}_{>0}$ we have

[TABLE]

where $G_{r}(s;x,\varphi)$ and $H_{r}(s;x,\varphi)$ are given by

[TABLE]

respectively. Here $(3/2-\sigma)$ denotes $\left\{3/2-\sigma+iv\mid v\in\mathbb{R}\right\}$.

Proof.

For $r\in\{0,1,\dots,m\}$ and $|v|\gg|t|$ let

[TABLE]

First we shall show that the integrand of (3.3) is holomorphic in $D_{\sigma}\setminus\{0\}$. From Lemma 2.1, $K_{\varphi}(w)(Q^{{2}/{d_{F}}}xe^{-i\pi{\rm sgn}(t)/2})^{{d_{F}w}/{2}}$ is holomorphic in $D_{\sigma}$. Since $F^{(m)}(w+s)$ has pole of order $p_{F}+m$ at most at $w=1-s$, we see that

[TABLE]

is holomorphic in $w\in D_{\sigma}$.

Here we shall consider the holomorphicity of gamma-factor of (3.3). In the case of ${\rm Re\>}\mu_{j}>\lambda_{j}/2$, since ${\rm Re}(\lambda_{j}(s+w)+\mu_{j})\geq-\lambda_{j}/2+{\rm Re\>}\mu_{j}>0$ for $w\in D_{\sigma}$ we see that $A_{j}(s+w)\Gamma(\lambda_{j}(s+w)+\mu_{j})=\Gamma(\lambda_{j}(s+w)+\mu_{j})$ is holomorphic in $D_{\sigma}$. On the other hand, in the case of ${\rm Re\;}\mu_{j}\leq\lambda_{j}/2$, we have ${\rm Re}(\lambda_{j}(s+w)+\mu_{j}+[\lambda_{j}/2-{\rm Re\;}\mu_{j}]+1)=1-\{\lambda_{j}/2-{\rm Re\;}\mu_{j}\}>0$ for $w\in D_{\sigma}$. Hence the functional equation for $\Gamma(s)$ implies that

[TABLE]

is holomorphic in $D_{\sigma}$.

Next we consider the existence of pole for $(\chi_{F}^{(m-r)}/\chi_{F})(s+w)$ in $D_{\sigma}$. In the case of ${\rm Re\;}\mu_{j}>\lambda_{j}/2$, since

[TABLE]

for $n\in\mathbb{Z}_{\geq 0}$, we see that $(\chi_{F}^{(m-r)}/\chi_{F})(s+w)$ does not have pole in $D_{\sigma}$. On the other hand, in the case of ${\rm Re\;}\mu_{j}\leq\lambda_{j}/2$, since

[TABLE]

from Lemma 2.4 the points $w=-s-(\mu_{j}+n)/\lambda_{j},\;1-s+(\overline{\mu_{j}}+n)/\lambda_{j}$ for $n\in\mathbb{Z}_{\geq 0}\cap\mathbb{Z}_{\leq[\lambda_{j}/2-{\rm Re\;}\mu_{j}]}$ are pole of order $(m-r)$ for $(\chi_{F}^{(m-r)}/\chi_{F})(s+w)$ in $D_{\sigma}$. Therefore

[TABLE]

is holomorphic in $D_{\sigma}$. Combining (3.4)–(3.6) we find that the integrand of (3.2) is holomorphic $w\in D_{\sigma}\setminus\{0\}$.

Next we shall show $I_{r}(v)\to 0$ when $v\to\infty$. Trivial estimate gives

[TABLE]

where $f_{F}=2\sum_{j=1}^{q}\max\{0,[\lambda_{j}/2-{\rm Re\>}\mu_{j}]\}$. To obtain an estimate of $F^{(r)}(s+w)$ in $w\in D_{\sigma}$, we shall consider estimates of $\chi_{F}(s)$ and $(\chi_{F}^{(r)}/\chi_{F})(s)$. By the same method of (2.16) and $it=|t|e^{i\frac{\pi}{2}{\rm sgn}(t)}$, we have

[TABLE]

where $\theta_{j}(t)=2\lambda_{j}+{\rm sgn}(t)\cdot(-\lambda_{j}-2{\rm Re\>}\mu_{j}+1)\pi/2-\log(\lambda_{j}|t|)^{2(\lambda_{j}+{\rm Im\>}\mu_{j})}$. Hence we obtain

[TABLE]

where $\theta_{F}(t)=d_{F}+{\rm sgn}(t)\cdot(-d_{F}/2-e_{F}+q)\pi/2-\log(C_{F}^{\prime}|t|^{d_{F}+e_{F}})$ and $C_{F}^{\prime}=(Q\prod_{j=1}^{q}\lambda_{j}^{\lambda_{j}+{\rm Im\>}\mu_{j}})^{2}$. Combining (1.2), (3.9) and Lemma 2.4 we have

[TABLE]

for $s\in\mathbb{C}:\sigma\in\mathbb{R}_{<0},|t|\gg 1$. Phragmén-Lindelöf theorem implies

[TABLE]

uniformly for $u\in[-1/2-\sigma,3/2-\sigma]$. By (3.7), (3.10), Lemmas 2.4, 2.6 we get

[TABLE]

when $|v|\gg|t|$. Choosing $l\in\mathbb{Z}_{>M_{F}}$ we find that $I_{r}(v)\to 0$ when $v\to\infty$, where $M_{F}=3d_{F}/4+(e_{F}-q)/2+2(p_{F}+m)+(m+1)f_{F}$. Cauchy’s integral theorem gives

[TABLE]

Here the first term of right-hand side of (3.11) is

[TABLE]

Since (1.3) implies

[TABLE]

combining this formula and (1.2) we obtain

[TABLE]

By replacing $w\mapsto-w$ and using this formula, (2.1) and $\overline{g_{F}}(1-s)=g_{F}(s)$, the second term of right-hand side of (3.11) is

[TABLE]

Therefore, from (3.11) and (3.13) Proposition 3.1 is obtained. ∎

Next applying residue theorem to the functions ${G}_{r}(s;x,\varphi)$ and ${H}_{r}(s;x,\varphi)$, then these functions are approximated as follows:

Proposition 3.2**.**

For any $F\in\mathcal{S}$, $m\in\mathbb{Z}_{\geq 0}$, $r\in\left\{0,\dots,m\right\}$, $s=\sigma+it:\sigma\in[0,1],|t|\gg 1$, $\varphi\in\mathcal{R}$, $l\in\mathbb{Z}_{>M_{F}}$ and $x,y\in\mathbb{R}_{>0}:\sqrt{C_{F}}(x|t|)^{\frac{d_{F}}{2}}=y$, we have

[TABLE]

where $M_{F}$ is some positive constant, and ${{\gamma}_{j,r}}(s;\rho)$, ${{\delta}_{j,r}}(s;\rho)$ are given by (1.5), (1.6) respectively.

Proof.

In order to show (3.14), we use (2.2) and write ${F^{(r)}}(s)=(\sum_{n\leq\rho y}+\sum_{n>\rho y})\times\times{a_{F}(n)}(-\log n)^{r}n^{-s}$ for ${\rm Re\;}s>1$ and $\rho\in\mathbb{R}_{>0}$. Then

[TABLE]

where $I_{1},I_{2}$ are given by

[TABLE]

To approximate $I_{1}$ and $I_{2}$, we define $L_{\pm j}$, $C_{j}$ ($j=1,2$) as

[TABLE]

respectively, where $\sigma_{1}=-1/2-\sigma$ and $\sigma_{2}=3/2-\sigma$. The residue theorem gives

[TABLE]

where $I_{1}^{\prime},I_{2}^{\prime}$ are (3.17) replaced by $L_{-1}+C_{1}+L_{+1}$, $L_{-2}+C_{2}+L_{+2}$ respectively, and ${\rm Res\;}(I_{1},\mathcal{F})$ is the sum of residue for the integrand of (3.17) in $\mathcal{F}$. Here by using the result

[TABLE]

for $\mu\in\mathbb{R}_{\geq 0}$ (see p.337 of [4]) and the residue theorem, ${\rm Res\;}(I_{1},\mathcal{F})$ is written as

[TABLE]

Next we shall estimate $I_{1}^{\prime}$ and $I_{2}^{\prime}$ as the error term of approximate functional equation. Partial summation and the assumption (1.1) give

[TABLE]

for ${\rm Re}(s+w)\leq-1/2$ and

[TABLE]

for ${\rm Re}(s+w)\geq 3/2$. From Lemma 2.4 and 2.6 we get the following estimate:

[TABLE]

Trivial estimate gives

[TABLE]

Hence by combining (3.21)–(3.24), $I_{1}^{\prime}$ is estimated as

[TABLE]

under the condition $\sqrt{C_{F}}(x|t|)^{\frac{d_{F}}{2}}=y$, where the following estimate was used:

[TABLE]

Since $1\ll 1+|t+v|\ll|t|$ when $v\in[-2|t|,-|t|/2]\cup[|t|/2,2|t|]$ and

[TABLE]

$J_{j}$ ($j=1,2,3$) were estimated as

[TABLE]

where $l$ was chosen as $l\in\mathbb{Z}_{\geq 2\max\{0,d_{F}-M_{F}\}}$. By the same discussion of estimate of $I_{1}^{\prime}$, the estimate of $I_{2}^{\prime}$ is obtained:

[TABLE]

Therefore combining (3.16)–(3.20), (3.25)–(3.26) we obtain (3.14). Since we can obtain (3.15) from the same discussion in the above, Proposition 3.2 is showed. ∎

Finally, for any $x\in\mathbb{R}_{>0}$ we choose parameters $y_{1},y_{2}\in\mathbb{R}_{>0}$ such that $\sqrt{C_{F}}(x|t|)^{\frac{d_{F}}{2}}\\ =y_{1}$ and $\sqrt{C_{F}}(x^{-1}|t|)^{\frac{d_{F}}{2}}=y_{2}$, that is, $y_{1}y_{2}=C_{F}|t|^{d_{F}}$. By combining Propositions 3.1, 3.2 and using (3.9), $F^{(m)}(s)$ is approximated as

[TABLE]

Using Lemma 2.7 and dividing sums of $j\in\mathbb{Z}_{\geq 0}$ to term of $j=0$ and sum of $j\in\mathbb{Z}_{\geq 1}$, we complete proof of Theorem 1.1.

4 Proof of Theorem 1.2

In order to prove Theorem 1.2 from the approximate functional equation containing characteristic functions, we use Lemma 2.8. For any functions $X:[0,\infty)\to\mathbb{R}$, we define $M_{X}(s)$ to

[TABLE]

Replacing $\varphi\mapsto\varphi_{\alpha}\;(\alpha\in\mathbb{R}_{\geq 0})$ in Theorem 1.1 we can write

[TABLE]

Here $M_{\xi}(s)$ and $M_{\varphi_{\alpha}-\xi}(s)+R_{\varphi_{\alpha}}(s)$ are written as

[TABLE]

and

[TABLE]

respectively, where $E(s)$, $S_{0}(\rho)$ and $T_{r}(\rho)$ are given by

[TABLE]

Since $S_{0}(\rho)=0$ and $T_{m-r}(\rho)=0$ for $\rho\in[0,(1+|t|^{-\alpha})^{-1}]\cup[1+|t|^{-\alpha},\infty)$ by Lemmas 2.7 and 2.8, we have

[TABLE]

for any $\alpha\in\mathbb{R}_{\geq 0}$, and

[TABLE]

for $\rho\in[(1+|t|^{-\alpha})^{-1},1+|t|^{-\alpha}]$ under the condition $\alpha\in[0,1/2]$. Now we choose $\alpha=1/2-\varepsilon$ and $l\in\mathbb{Z}_{\geq 1/(2\varepsilon)}$. By the condition (d) of Selberg class and the estimates (3.9), (4.3)–(4.5) and $(1+|t|^{-\alpha})y_{1}-(1+|t|^{-\alpha})^{-1}y_{1}\leq 2|t|^{-\alpha}y_{1}$, $M_{\varphi_{\alpha}-\xi}(s)+R_{\varphi_{\alpha}}(s)$ is estimated as

[TABLE]

Hence combining (4.1), (4.2) and (4.6) we complete the proof of Theorem 1.2.