2.1 The parameters (Convention 1.1)

We fix $\epsilon _{0}:=1/1000$ , and assume that $\alpha \in \mathfrak {a}_{\mathbb {C}}:= \{ (\alpha _{1},\alpha _{2}, \alpha _{3})\in \mathbb {C}^{3}: \alpha _{1}+\alpha _{2}+\alpha _{3}=0 \}$ , the components $\alpha _{i}$ ’s are distinct and satisfy $ |\alpha _{1}|, |\alpha _{2}| < \epsilon _{0}/1000$ . We let $\operatorname {\mathrm {\boldsymbol {\alpha }}} :=\max _{1\le i\le 3} \, |\operatorname {\mathrm {Re}} \, \alpha _{i}| \,(<\epsilon _{0}/500)$ .

2.2 Test functions (Regularity 1.2)

We fix $\eta>40$ . Our test function H lies in the class $ \mathcal {C}_{\eta }$ . The function $h=H^{\flat }$ is the Kontorovich–Lebedev inversion of H as defined in (3.13).

2.3 Analysis

We use the same symbol to denote a function and its analytic continuation. We denote the real part of s by $\sigma $ . We write ‘ $f(y) \ll _{A} g(y)$ ’ (or ‘ $g(y)\gg _{A} f(y)$ ’) if there exists a constant $C>0$ , depending on A, such that $|f(y)|\le C|g(y)|$ for sufficiently large y.

The Euler $\Gamma $ -function is defined by

(2.1) $$ \begin{align} \Gamma(s) \, := \, \int_{0}^{\infty} e^{-x} x^{s} \, d^{\times} x \end{align} $$

on $\operatorname {\mathrm {Re}} s>0$ . It admits a meromorphic continuation to $\mathbb {C}$ . Let $\Gamma _{\mathbb {R}}(s):= \pi ^{-s/2}\Gamma (s/2)$ for $s\in \mathbb {C}$ . We use both $\Gamma (s)$ and $\Gamma _{\mathbb {R}}(s)$ as the former is common in our references, but the latter simplifies our formulae on several occasions.

The contours of the Barnes integrals

(2.2) $$ \begin{align} \int_{-i\infty}^{i\infty} \, \cdots \, \int_{-i\infty}^{i\infty} \ (\cdots) \ \frac{ds_{1}}{2\pi i} \, \cdots \, \frac{ds_{k}}{2\pi i} \end{align} $$

are chosen so that they pass to the right of all poles of the gamma functions of the form $\Gamma (s_{i}+ a)$ , and to the left of all poles of those of the form $\Gamma (a -s_{i})$ .

Let $\Lambda (s) := \Gamma _{\mathbb {R}}(s)\zeta (s)$ denote the completed Riemann zeta function. It admits a holomorphic continuation to $\mathbb {C}$ except for simple poles at $s=0,1$ , and satisfies the functional equation

(2.3) $$ \begin{align} \Lambda(s) = \Lambda(1-s) \end{align} $$

for any $s\in \mathbb {C}$ .

The K-Bessel function is defined by

(2.4) $$ \begin{align} K_{\mu}(y) \, := \, \frac{1}{2}\int_{0}^{\infty} \exp\Big(-\frac{y}{2}(t+t^{-1})\Big) t^{-\mu} \, d^{\times} t \end{align} $$

for any $y>0$ and $\mu \in \mathbb {C}$ .

The Gauss $_{2}F_{1}$ hypergeometric function is defined by

(2.5) $$ \begin{align} \ \mathbf{F}\left(\begin{matrix} a,\; b \\ c \end{matrix} \Bigm|z \right) \ := \ \frac{\Gamma(c)}{\Gamma(a) \Gamma(b) }\, \sum_{n=0}^{\infty} \ \frac{\Gamma(a+n) \Gamma(b+n) }{\Gamma(c+n) } \frac{z^{n}}{n!} \end{align} $$

for $a,b,c \not \in \mathbb {Z}_{\le 0}$ . The series converges absolutely on $|z|<1$ , and on $|z|=1$ if $\operatorname {\mathrm {Re}} (c-a-b)>0$ . Furthermore, it has an analytic continuation to $\mathbb {C} - [1, \infty )$ as a function of z.

Let $h: (0,\infty )\rightarrow \mathbb {C}$ be a continuous function. Its Mellin transform and inversion formula are given, respectively, by

(2.6) $$ \begin{align} \widetilde{h}(s) \, := \, \int_{0}^{\infty} h(y)y^{s} \, d^{\times} y \hspace{15pt} \text{and} \hspace{15pt} h(y) \, = \, \int_{(\sigma)} \, \widetilde{h}(s)y^{-s} \, \frac{ds}{2\pi i}, \end{align} $$

provided that both integrals converge absolutely.

2.4 Groups

The Weyl group $W_{3}$ of $\mathrm {GL}_{3}$ is be described in (3.29). For $n \in \{2,3\}$ , let $\mathrm {G}_{n}:= \mathrm {GL}_{n}(\mathbb {R})$ , $\overline {\mathrm {G}}_{n}:= \mathrm {G}_{n}/\mathbb {R}_{>0}^{\times }$ , $\Gamma _{n}:=\mathrm {SL}_{n}(\mathbb {Z})$ the full modular group; $\mathrm {K}_{n}:=\mathrm {O}_{n}$ the maximal compact subgroup of $\mathrm {G}_{n}$ ; $\mathrm {U}_{n}$ the subgroup of upper-triangular unipotent matrices in $\mathrm {G}_{n}$ ; $\mathrm {N}_{ij}$ the one-parameter unipotent subgroup attached to the $(i,j)$ -th entry, where $i\neq j$ .

The following quotients occur frequently in this work: $[\mathrm {U}_{n}]:= (\mathrm {U}_{n}\cap \Gamma _{n})\setminus \mathrm {U}_{n}$ , $[\mathrm {N}_{ij}]:= (\mathrm {N}_{ij}\cap \Gamma _{n})\setminus \mathrm {N}_{ij}$ , $[\mathrm {G}_{n}]:= \Gamma _{n}\setminus \mathrm {G}_{n}$ , $[\overline {\mathrm {G}}_{n}]:= \Gamma _{n}\setminus \overline {\mathrm {G}}_{n}$ , $[\Gamma _{n}]:= (\mathrm {U}_{n}\cap \Gamma _{n})\setminus \Gamma _{n}$ , and $\mathfrak {h}^n :=\mathrm {G}_{n}/\mathrm {K}_{n}$ . Let $\mathrm {Y}^{+}$ be the group of matrices of the form $\mathbf {y}:= \mathrm {diag}(y_{0}y_{1}, y_{0},1)$ with $y_{0}, y_{1}>0$ . We have the following identifications:

We write $\mathbf {u}:= (u_{ij})_{1\le i,j\le 3} \in \mathrm {U}_{3}$ ; $\mathbf {n}, \, \mathbf {n}_{ij}(x) \in \mathrm {N}_{ij}$ with $x\in \mathbb {R}$ being the $(i,j)$ -th entry of $\mathbf {n}_{ij}(x)$ ; $\mathbf {y}(y_{0}, y_{1})= \mathrm {diag}(y_{0}y_{1}, y_{0},1)$ and $f(\mathbf {y})=f(y_{0}, y_{1})$ for $y_{0},\, y_{1}>0$ ; and for $\gamma \in \Gamma _{2}$ and $g\in \mathrm {G}_{3}$ , we set

(2.7) $$ \begin{align} \gamma g \, := \, \mathrm{diag}(\gamma, 1)g. \end{align} $$

2.5 Additive characters

Let $e(x):=e^{2\pi i x}$ for $x\in \mathbb {R}$ . We label the characters of $[\mathrm {U}_{3}]$ and $[\mathrm {N}_{ij}]$ by $\psi _{(m_{1},m_{2})}(\mathbf {u}):= e(m_{1} u_{23}+m_{2} u_{12})$ and $\psi _{m}(\mathbf {n}):= e(m n_{ij})$ , respectively, where $m, m_{1}, m_{2}\in \mathbb {Z}$ , $\mathbf {u}\in \mathrm {U}_{3}$ , and $\mathbf {n} \in \mathrm {N}_{ij}$ . We also write $\psi _{\pm }= \psi _{(1, \, \pm 1)}$ and $\psi =\psi _{+}$ .

2.6 Measures

We use the shorthand $d^{*}\mathbf {y} = (y_{0}y_{1}^2)^{-1}\,dy_{0} \,dy_{1}$ . The normalized spherical Whittaker–Plancherel measure for $\mathrm {G}_{2}$ is given by

(2.8) $$ \begin{align} d^{W}\mu \, := \, |\Gamma_{\mathbb{R}}(\mu)|^{-2} \, \frac{d\mu}{4\pi i} \hspace{20pt} (\mu \, \in \, i\mathbb{R}). \end{align} $$

The invariant measure on $\mathrm {G}_{2}$ can be described by the Iwasawa decomposition as follows:

where $x\in \mathbb {R}$ , $y, z>0$ , $k\in \mathrm {K}_{2}$ , and $\mathrm {meas}(\mathrm {K}_{2})=1$ . The Petersson inner product is defined by

(2.9) $$ \begin{align} \left\langle \, \phi_{1},\, \phi_{2} \, \right\rangle \ := \ \int_{[\overline{\mathrm{G}}_{2}]} \ \phi_{1}(g) \,\overline{\phi_{2}(g)} \ dg \end{align} $$

for smooth functions $\phi _{1}, \phi _{2}$ on $[\overline {\mathrm {G}}_{2}]$ . If $\phi _{1}, \phi _{2}$ are spherical (i.e., $\mathrm {K}_{2}$ -invariant), the domain of (2.9) can of course be replaced by $\Gamma _{2}\setminus \mathfrak {h}^2$ .

2.7 Automorphic objects

In this article, our Whittaker and automorphic functions are defined on $\overline {\mathrm {G}}_{n}$ and are spherical (or right $\mathrm {K}_{n}$ -invariant). Equivalently, they are functions on the locally symmetric space $\mathfrak {h}^n$ . For brevity, we may omit the term ‘spherical’ in what follows.

Our automorphic forms are Hecke–Maass forms for $\Gamma _{n}$ , i.e., they are smooth functions of moderate growth on $\overline {\mathrm {G}}_{n}$ that are right $\mathrm {K}_{n}$ -invariant, left $\Gamma _{n}$ -invariant, and are joint eigenfunctions to the commutative ring of invariant differential operators on $\mathfrak {h}^n$ (denoted by $\mathcal {D}_{n}$ ) and the ring of Hecke operators. Their first Fourier coefficients are normalized to be $1$ . They are either cuspidal or Eisenstein.

We use $E_{\min }^{(3)}$ to denote the minimal parabolic Eisenstein series for $\Gamma _{3}$ , which we define in (3.28). From Section 3.3.3 onward, we use $\Phi $ exclusively for the completed version of $E_{\min }^{(3)}$ as given in (3.33).

The Fourier–Whittaker coefficients and periods of an automorphic form $\phi $ of $\Gamma _{2}$ are denoted, respectively, by $\mathcal {B}_{\phi }(a)$ and $\mathcal {W}_{a}(g; \, \phi )$ ; those of an automorphic form $\Phi $ of $\Gamma _{3}$ are denoted by $\mathcal {B}_{\Phi }(m_{1}, m_{2})$ and $\mathcal {W}_{(m_{1},m_{2})}(g; \, \Phi )$ . We write $W_{\mu }(g)$ and $W_{\alpha }(g)$ for the standard Whittaker functions for $\overline {\mathrm {G}}_{2}$ and $\overline {\mathrm {G}}_{3}$ , respectively, and $W_{\alpha ,\, w}^{(0,1)}(\mathbf {y})$ and $W_{\alpha , \, w}^{(1,0)}(\mathbf {y})$ for the degenerate Whittaker functions for $\overline {\mathrm {G}}_{3}$ . There is no ambiguity as the group to which g belongs is clear from the context. We use $\mu $ and $\alpha $ to denote the spectral parameters for, respectively, the automorphic forms (or Whittaker functions) of $\Gamma _{2}$ and $\Gamma _{3}$ . In our case, $\mu $ is always purely imaginary because of (2.8) and the known Selberg’s conjecture for $\Gamma _{2}$ ([Reference GoldfeldGol06, Theorem 3.7.2]).

The following unipotent period for $\mathrm {G}_{3}$ plays an important role in our work:

(2.10) $$ \begin{align} \mathcal{P}_{\psi}(g; \, F) \, := \, \int_{[\mathrm{N}_{12}]} F( \mathbf{n} g) \overline{\psi(\mathbf{n})} \ d\mathbf{n}, \end{align} $$

where $F: \mathrm {N}_{12}(\mathbb {Z})\setminus \mathrm {G}_{3}\rightarrow \mathbb {C}$ is a smooth function. It is customary to write for $g\in \mathrm {G}_{2}$ and a function F defined on $\mathrm {G}_{3}$ . Suppose F and $\phi $ are smooth functions on $[\mathrm {G}_{2}]$ . We define

(2.11) $$ \begin{align} \mathcal{I}(s; \, \phi, \, F) \, := \, \int_{[\mathrm{G}_{2}]} \, \phi(g)F(g) |\det g|^{s-\frac{1}{2}} \ dg \hspace{15pt} (s \, \in \, \mathbb{C}) \end{align} $$

whenever in the integral converges absolutely. In our previous works [Reference KwanKwa24, Reference KwanKwa25], the integral (2.11) was instead denoted by $(\phi ,\ (\mathbb {P}_{2}^{3}\widetilde {F})\, |\det *|^{\overline {s}-\frac {1}{2}}) _{\Gamma _{2}\setminus \mathrm {GL}_{2}(\mathbb {R})}$ , where $\widetilde {F}(g):= F(^{t}g^{-1})$ for $g\in \mathrm {G}_{3}$ .

3.1 Barnes’ identities

Lemma 3.1. For $a,b,c,d,e,f \in \mathbb {C}$ with $f=a+b+c+d+e$ , we have

(3.1) $$ \begin{align} &\int_{-i\infty}^{i\infty} \frac{\Gamma(w+a) \Gamma(w+b)\Gamma(w+c) \Gamma(d-w) \Gamma(e-w)}{\Gamma(w+f)} \frac{dw}{2\pi i} \nonumber\\ & \quad = \frac{\Gamma(d+a) \Gamma(d+b) \Gamma(d+c) \Gamma(e+a) \Gamma(e+b)\Gamma(e+c)}{\Gamma(f-a) \Gamma(f-b) \Gamma(f-c)}. \end{align} $$

Lemma 3.2. For $x>0$ , we have

(3.2) $$ \begin{align} \mathbf{F}\left(\begin{matrix} a,\; b \\ c \end{matrix} \Bigm|-x^{-1} \right) \, = \, \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)} \, \int_{-i\infty}^{i\infty} \, \frac{\Gamma(w+a)\Gamma(w+b)\Gamma(-w)}{\Gamma(w+c)} x^{-w} \, \frac{dw}{2\pi i}. \end{align} $$

3.2 Whittaker functions and transforms

It is well known that the standard Whittaker function for $\overline {\mathrm {G}}_{2}$ is given in terms of the K-Bessel function (see (2.4)):

(3.3) $$ \begin{align} W_{\mu}(g) \ := \ 2\sqrt{y} K_{\mu}(2\pi y) e(x) \end{align} $$

for $\mu \in \mathbb {C}$ and . We have the following Mellin integral formula for $W_{\mu }$ :

Lemma 3.3. For $\operatorname {\mathrm {Re}} w> -1/2+|\operatorname {\mathrm {Re}} \mu |$ , we have

(3.4) $$ \begin{align} \int_{0}^{\infty} W_{\mu}(y) y^{w} \ d^{\times} y = \frac{1}{2} \ \Gamma_{\mathbb{R}}(w+1/2+\mu)\Gamma_{\mathbb{R}}(w+1/2-\mu). \end{align} $$

For the group $\overline {\mathrm {G}}_{3}$ , we first introduce the power function

(3.5) $$ \begin{align} I_{\alpha}(y_{0}, y_{1}) \, = \, I_{\alpha}(\mathbf{y}) \, := \, y_{0}^{1-\alpha_{3}} y_{1}^{1+\alpha_{1}} \end{align} $$

for $\mathbf {y}\in \mathrm {Y}^{+}$ and $\alpha \in \mathfrak {a}_{\mathbb {C}}$ .Footnote ³ This can be regarded as a function defined on $\mathrm {U}_{3}\setminus \mathfrak {h}^{3}$ by the Iwasawa decomposition. The standard Whittaker (or Jacquet–Whittaker) function for $\overline {\mathrm {G}}_{3}$ is defined by

(3.6) $$ \begin{align} W_{\alpha}^{\pm}(g) \, := \, \prod_{1\le j<k\le 3} \Gamma_{\mathbb{R}}(1+\alpha_{j}-\alpha_{k}) \ \int_{\mathrm{U}_{3}} \ I_{\alpha}(w_{\ell} \mathbf{u} g) \, \overline{\psi_{\pm}(\mathbf{u})} \, d\mathbf{u} \hspace{20pt} (g \, \in \, \mathrm{G}_{3}). \end{align} $$

The integral in (3.6) converges absolutely whenever $\operatorname {\mathrm {Re}} (\alpha _{1}-\alpha _{2})>0$ and $\operatorname {\mathrm {Re}} (\alpha _{2}-\alpha _{3})>0$ , and $W_{\alpha }^{\pm }$ admits a holomorphic continuation to $\mathfrak {a}_{\mathbb {C}}$ ; see [Reference GoldfeldGol06, Chapter 5.5]. Several normalizations of the standard Whittaker function are present in the literature. We opt for the one such that the functional equation $W_{\alpha }^{\pm }(g)=W_{\alpha ^{w}}^{\pm }(g)$ holds for any $w\in W_{3}$ and $g\in \mathrm {G}_{3}$ .

Moreover, we have $W_{\alpha }^{+}(\mathbf {y})=W_{\alpha }^{-}(\mathbf {y})$ , and for brevity, we denote this quantity by $W_{\alpha }(\mathbf {y})=W_{\alpha }(y_{0}, y_{1})$ . Since $I_{\alpha }$ is a joint eigenfunction of $\mathcal {D}_{3}$ ,Footnote ⁴ so is $W_{\alpha }$ . Finally, we have the following Mellin–Barnes integral formula (also known as the Vinogradov–Takhtadzhyan formula) for $W_{\alpha }(\mathbf {y})$ :

Lemma 3.4. Let $\operatorname {\mathrm {\boldsymbol {\alpha }}} :=\max _{1\le i\le 3} \, |\operatorname {\mathrm {Re}} \, \alpha _{i}|$ . For any $\sigma _{0},\, \sigma _{1}> \operatorname {\mathrm {\boldsymbol {\alpha }}}$ and $\mathbf {y}\in \mathrm {Y}^{+}$ , we have

(3.7) $$ \begin{align} W_{-\alpha}(\mathbf{y}) = \frac{1}{4} \ \int_{(\sigma_{0})}\int_{(\sigma_{1})} G_{\alpha}(s_{0},s_{1}) y_{0} ^{1-s_{0}} y_{1}^{1-s_{1}} \ \frac{ds_{0}}{2\pi i } \frac{ds_{1}}{2\pi i }, \end{align} $$

where

(3.8) $$ \begin{align} G_{\alpha}(s_{0},s_{1}) \ := \ \frac{\prod_{i=1}^{3} \Gamma_{\mathbb{R}}\left( s_{0}+\alpha_{i}\right) \Gamma_{\mathbb{R}}\left( s_{1}-\alpha_{i}\right)}{\Gamma_{\mathbb{R}}(s_{0}+s_{1})}. \end{align} $$

Corollary 3.6. Under Convention 1.1, we have, for any $0< a_{i}< 1-\operatorname {\mathrm {\boldsymbol {\alpha }}}$ and $ A_{i} \,> \, 0$ ( $i=1,2$ ), that

(3.9) $$ \begin{align} |W_{-\alpha}(\mathbf{y})| \ \ll_{A_{0}, \, A_{1}} \ y_{0}^{a_{0}}(1+y_{0})^{-A_{0}} y_{1}^{a_{1}} (1+y_{1})^{-A_{1}}, \hspace{20pt} y_{0}, \, y_{1} \> \ 0. \end{align} $$

We will need the explicit evaluation of the archimedean $\text {GL}(3)\times \text {GL}(2)$ Rankin–Selberg integral, which is a consequence of the second Barnes Lemma. This calculation also verifies the sign convention of (3.7).

Lemma 3.7. For $\sigma> \operatorname {\mathrm {\boldsymbol {\alpha }}} + |\operatorname {\mathrm {Re}} \mu |$ , we have

(3.10) $$ \begin{align} \mathcal{Z}_{\infty}(s; \, W_{\mu},\, W_{-\alpha}) & := \int_{0}^{\infty} \int_{0}^{\infty} \, W_{\mu}(y_{1})W_{-\alpha}(y_{0}, y_{1}) (y_{0}^2 y_{1})^{s-\frac{1}{2}} \ d^{*}\mathbf{y} \nonumber\\ &= \frac{1}{4} \, \prod_{\pm} \prod_{k=1}^{3} \, \Gamma_{\mathbb{R}}\left(s\pm \mu-\alpha_{k}\right). \end{align} $$

Proof. This was proved in [Reference BumpBum88]. To clarify our conventions, we provide a short proof here. Take $\sigma _{1}\in (\operatorname {\mathrm {\boldsymbol {\alpha }}} + |\operatorname {\mathrm {Re}} \mu |, \, \sigma )$ . From Lemma 3.4 and Mellin inversion, we have, for $\operatorname {\mathrm {Re}} s_{0}> \operatorname {\mathrm {\boldsymbol {\alpha }}}$ , that

$$ \begin{align*} \int_{0}^{\infty} \ (y_{0}y_{1})^{-1}W_{-\alpha}(\mathbf{y})\, y_{0}^{s_{0}} \ d^{\times} y_{0} = \int_{(\sigma_{1})} \ \frac{G_{\alpha}(s_{0}, s_{1})}{4} y_{1}^{-s_{1}} \ \frac{ds_{1}}{2\pi i}. \nonumber \end{align*} $$

Plugging this into the double integral in (3.10) and interchanging the order of integration, we have

(3.11) $$ \begin{align} \mathcal{Z}_{\infty}(s; \, W_{\mu},\, W_{-\alpha}) \ =\ \int_{(\sigma_{1})} \ \frac{G_{\alpha}(2s, s_{1})}{4} \, \int_{0}^{\infty} \ W_{\mu}(y_{1}) y_{1}^{s-\frac{1}{2}-s_{1}} \ d^{\times} y_{1} \ \frac{ds_{1}}{2\pi i}. \end{align} $$

The innermost integral can be computed by (3.4). Together with (3.8), it follows that

$$ \begin{align*} & \mathcal{Z}_{\infty}(s; \, W_{\mu},\, W_{-\alpha}) \\ &\quad = \frac{\pi^{-3s}}{8}\, \prod_{k=1}^{3} \Gamma\left( s+\frac{\alpha_{k}}{2}\right) \, \int_{(\sigma_{1})} \, \frac{\prod_{k=1}^{3} \Gamma\left( \frac{s_{1}-\alpha_{k}}{2}\right)\cdot \Gamma\left(\frac{s+\mu-s_{1}}{2}\right) \Gamma\left(\frac{s-\mu-s_{1}}{2}\right)}{\Gamma\left( \frac{s_{1}}{2}+s\right)} \, \frac{ds_{1}}{2\pi i}. \end{align*} $$

The desired result immediately follows from a change of variable $s_{1}\to 2 s_{1}$ and (3.1) with the choice of the parameters $ \left ( a, b,c ;d; e\right ) := \left ( -\frac {\alpha _{1}}{2}, \ -\frac {\alpha _{2}}{2}, \ -\frac {\alpha _{3}}{2}; \ \frac {s+\mu }{2}, \ \frac {s-\mu }{2}\right )$ .

The following pair of integral transforms plays a central role in the analysis of this article.

Definition 3.8. Let $h: (0, \infty ) \rightarrow \mathbb {C}$ and $H: i\mathbb {R} \rightarrow \mathbb {C}$ be measurable functions with $H(\mu )=H(-\mu )$ . Let $W_{\mu }$ be given by (3.3). Then the Kontorovich–Lebedev transform of h is defined by

(3.12) $$ \begin{align} h^{\#}(\mu) \, := \, \int_{0}^{\infty} h(y) W_{\mu}(y) \ \frac{dy}{y^2}, \end{align} $$

whereas its inverse transform is defined by

(3.13) $$ \begin{align} H^{\flat}(y) \, = \, \int_{(0)} \, H(\mu) W_{\mu}(y) \ d^{W}\mu, \hspace{15pt} \text{where} \hspace{10pt} d^{W}\mu \, := \, |\Gamma(\mu)|^{-2} \, \frac{d\mu}{4\pi i}, \end{align} $$

provided that the integrals above converge absolutely.

Lemma 3.9. Suppose $H\in \mathcal {C}_{\eta }$ . The integral in (3.13) that defines $H^{\flat }$ converges absolutely, and

(3.14) $$ \begin{align} H^{\flat}(y) \ \ll \ (y+1/y)^{\eta} \end{align} $$

for any $y>0$ . Moreover, the Whittaker–Plancherel formula holds for any $|\operatorname {\mathrm {Re}} \mu |< 2\eta $ :

(3.15) $$ \begin{align} (H^{\flat})^{\#}(\mu) = H(\mu). \end{align} $$

Lemma 3.11. Suppose $H\in \mathcal {C}_{\eta }$ and $h:= H^{\flat }$ . On the strip $-1/2< \operatorname {\mathrm {Re}} w < \eta $ , we have

(3.16) $$ \begin{align} \widetilde{h}(w) = \frac{1}{2} \int_{(0)} \ H(\mu) \Gamma_{\mathbb{R}}(w+1/2+\mu)\Gamma_{\mathbb{R}}(w+1/2-\mu) \ d^{W}\mu. \end{align} $$

3.3 Automorphic preliminaries

3.3.1 $\mathrm {PGL}(2)$

The invariant differential operator on $\mathfrak {h}^2$ is the hyperbolic Laplacian $\Delta := -y^2( \partial _{x}^2 +\partial _{y}^2)$ . An automorphic form $\phi : \mathfrak {h}^2 \rightarrow \mathbb {C}$ of $\Gamma _{2}=\mathrm {SL}_{2}(\mathbb {Z})$ satisfies $\Delta \phi = ( 1/4 -\mu ^2) \phi $ for some $\mu = \mu (\phi ) \in \mathbb {C}$ . The Fourier coefficient of $\phi $ , denoted by $\mathcal {B}_{\phi }(a)$ , is defined by

(3.17) $$ \begin{align} \mathcal{W}_{a}(y; \phi) \ := \ \int_{\mathbb{Z}\setminus \mathbb{R}} \ \phi\left[ \begin{pmatrix} 1 & x\\ & 1 \end{pmatrix} \begin{pmatrix} y & \\ & 1 \end{pmatrix}\right] e(-ax) \, dx = \frac{\mathcal{B}_{\phi}(a)}{\sqrt{|a|}} W_{\mu(\phi)}(|a|y) \end{align} $$

for any $a\in \mathbb {Z}-\{0 \}$ and $y>0$ .

Let $I_{\mu }(y):= y^{\mu +\frac {1}{2}}$ , which can be regarded as a function on $\mathrm {U}_{2}\setminus \mathfrak {h}^2$ . The Eisenstein series of $\Gamma _{2}$ is defined by

(3.18) $$ \begin{align} E(g; \mu) \ := \ \frac{1}{2} \ \sum_{\gamma\in [\Gamma_{2}]} \, I_{\mu}(\gamma g) \hspace{20pt} (g \,\in \, \mathrm{G}_{2}). \end{align} $$

The series (3.18) converges absolutely for $\operatorname {\mathrm {Re}} \mu>1/2$ and admits a meromorphic continuation to $\mathbb {C}$ ; see [Reference GoldfeldGol06, Chapter 3.1]. We have $\Delta E(*; \mu ) = ( 1/4-\mu ^2) E(*; \mu )$ , and the Fourier coefficients of $E(*;\mu )$ are explicitly given by

(3.19) $$ \begin{align} \mathcal{B}_{\mu}(a) = \frac{|a|^{\mu}\sigma_{-2\mu}(|a|)}{\Lambda(1+2\mu)}, \end{align} $$

where $\Lambda (s):= \pi ^{-s/2}\Gamma (s/2)\zeta (s)$ and $\sigma _{-2\mu }(|a|) \ := \ \sum _{d \mid a} d^{-2\mu }$ .

3.3.2 $\mathrm {PGL}(3)$

Suppose $\Phi : \mathfrak {h}^3 \rightarrow \mathbb {C}$ is an automorphic form of $\Gamma _{3}:= \mathrm {SL}_{3}(\mathbb {Z})$ . By the Harish-Chandra isomorphism, there exists $\alpha = \alpha (\Phi )\in \mathfrak {a}_{\mathbb {C}}$ such that for any $D\in \mathcal {D}_{3}$ , we have

$$ \begin{align*} D\Phi = \lambda_{\alpha}(D) \Phi \hspace{20pt} \text{ and } \hspace{20pt} DI_{\alpha} = \lambda_{\alpha}(D) I_{\alpha} \end{align*} $$

for some $\lambda _{\alpha }(D) \in \mathbb {C}$ , where $I_{\alpha }$ is given by (3.5). The triple $\alpha $ is said to be the spectral parameters of $\Phi $ . The spectral parameters of the dual form $\widetilde {\Phi }(g):= \Phi ( ^{t}g^{-1})$ are $-\alpha $ .

Definition 3.12. Let $(m_{1}, m_{2})\in \mathbb {Z}^{2}$ and $\Phi : \Gamma _{3}\setminus \mathfrak {h}^3 \rightarrow \mathbb {C}$ be an automorphic form. The $(m_{1}, m_{2})$ -th Fourier–Whittaker period of $\Phi $ refers to the integral

(3.20) $$ \begin{align} \mathcal{W}_{(m_{1}, m_{2})}(g; \, \Phi) \ &:= \ \int_{[\mathrm{U}_{3}]}\Phi(\mathbf{u} g) \overline{\psi_{(m_{1},m_{2})}(\mathbf{u})} \ d\mathbf{u} \hspace{60pt} (g\in \mathrm{G}_{3}). \end{align} $$

When $(m_{1}, m_{2})\in (\mathbb {Z}-\{0\})^{2}$ , the $(m_{1}, m_{2})$ -th Fourier coefficient of $\Phi $ refers to the complex number $\mathcal {B}_{\Phi }(m_{1}, m_{2})$ such that

(3.21) $$ \begin{align} \mathcal{W}_{(m_{1}, m_{2})}(\mathbf{y}; \, \Phi) = \frac{\mathcal{B}_{\Phi}(m_{1}, m_{2})}{|m_{1}m_{2}|} \ W_{\alpha(\Phi)}(|m_{1}|y_{0}, \, |m_{2}|y_{1}) \hspace{20pt} (\mathbf{y}\, \in \, \mathrm{Y}^{+}). \end{align} $$

Lemma 3.13. Let $\Phi : [\overline {\mathrm {G}}_{3}]\rightarrow \mathbb {C}$ be a smooth automorphic function. ThenFootnote ⁵

(3.22) $$ \begin{align} \Phi(g) = \mathcal{W}_{(0,0)}(g; \, \Phi) + \sum_{n_{2}\neq 0} \ \mathcal{W}_{(0, n_{2})}(g; \, \Phi) + \sum_{\gamma\in [\Gamma_{2}]} \ \sum_{n_{1}= 1}^{\infty} \ \mathcal{W}_{(n_{1},0)}(\gamma g; \, \Phi) + \Phi^{\mathrm{ND}}(g) \end{align} $$

for $g\in \mathrm {G}_{3}$ , where $\Phi ^{\mathrm {ND}}$ denotes the non-degenerate part of $\Phi $ :

(3.23) $$ \begin{align} \Phi^{\mathrm{ND}}(g) \ := \ \sum_{\gamma \in [\Gamma_{2}]} \ \sum_{m_{1}=1}^{\infty} \ \sum_{m_{2}\neq 0} \mathcal{W}_{(m_{1},m_{2})}( \gamma g; \, \Phi). \end{align} $$

Proof. See [Reference Ichino and YamanaIY15, Proposition 4.2].

The following identity is central to our method.

Proposition 3.14. For any smooth function $\Phi : [\overline {\mathrm {G}}_{3}]\rightarrow \mathbb {C}$ , we have, for any $g\in \mathrm {G}_{3}$ , that

(3.24) $$ \begin{align} \mathcal{P}_{\psi}(g; \, \Phi) \, := \, \int_{[\mathrm{N}_{12}]} \Phi(\mathbf{n} g) \overline{\psi(\mathbf{n})} \ d\mathbf{n} = \sum_{a_{0}\in \mathbb{Z}} \, \sum_{a_{1}\in \mathbb{Z}} \, \mathcal{W}_{(a_{1},1)}\big( \mathbf{n}_{32}(-a_{0})g;\, \Phi\big). \end{align} $$

Proof. See [Reference KwanKwa24, Proposition 6.1].

Remark 3.15. As explained in [Reference KwanKwa24, Section 4], it is equivalent to consider the unipotent period

(3.25) $$ \begin{align} \int_{[\mathrm{N}_{23}]} \, \Psi\left[\mathbf{n} \left( \begin{smallmatrix} y & & \\ & 1& \\ & & 1 \end{smallmatrix} \right)\right] \overline{\psi(\mathbf{n})} \, d\mathbf{n} \end{align} $$

for $\Psi := \rho (w_{\ell })\Phi $ . Recall the following construction used in the integral representation of the $\mathrm {GL}_{3}$ standard L-function (see [Reference GoldfeldGol06, Chapters 6.5 & 12.3]):

(3.26)

for any $g\in \mathrm {G}_{2}$ . Then the corresponding identity for (3.25) is

(3.27)

By (3.20) and $(\mathbb {P}_{1}^{3}\Psi )(g) \, = \, \sum _{n\neq 0} \, \mathcal {W}_{(1,n)}(g; \,\Psi )$ , upon inserting the second term on the right-hand side of (3.27) into our Fourier–Hecke period ((1.18) or (6.7)) and applying Mellin inversion, one obtains an integral moment of the product of the $\zeta $ -function (from the a-sum) and the L-function of $\Phi $ (from the n-sum; see (3.38)). The corresponding integral transform is given in (4.1). These steps were performed carefully in our previous works [Reference KwanKwa24, Reference KwanKwa25], and will be reviewed in Sections 6.5 and 6.7.

3.3.3 Eisenstein series for $\mathrm {PGL}(3)$

Explicit computations of the Fourier–Whittaker periods for the Eisenstein series of $\Gamma _{3}$ are required to realize the moment $ \mathfrak {M}_{-\alpha }^{(3)}(s; H)$ as a period, extract the main terms and recast them in a form predicted by the CFKRS conjectures, and carry out the analytic continuation argument of Section 6.

A standard reference is [Reference BumpBum84, Chapter VII]; see especially Theorem 7.2. For normalizations of the degenerate Whittaker functions of $\overline {\mathrm {G}}_{3}$ , readers should refer to [Reference BumpBum84, Chapter III]. It is also helpful to consult the reformulation by [Reference ButtcaneBut18, Section 4], which is more streamlined and up-to-date. Readers should beware that [Reference ButtcaneBut18] uses the ‘incomplete’ Whittaker functions, whereas both this article and [Reference BumpBum84] use the ‘completed’ ones.

Definition 3.16. The minimal parabolic Eisenstein series of $\Gamma _{3}$ is defined by

(3.28) $$ \begin{align} E_{\min}^{(3)}(g; \alpha) \ := \ \sum_{\gamma \in [\Gamma_{3}]} \ I_{\alpha}(\gamma g) \hspace{20pt} (g \,\in \, \mathrm{G}_{3}). \end{align} $$

The series (3.28) converges absolutely on $\big \{\alpha \in \mathfrak {a}_{\mathbb {C}} :\, \operatorname {\mathrm {Re}} (\alpha _{1}-\alpha _{2})> 1, \, \operatorname {\mathrm {Re}} (\alpha _{2}-\alpha _{3}) > 1\big \}$ (see [Reference ButtcaneBut18, Section 4.1]). Its meromorphic continuation to $\mathfrak {a}_{\mathbb {C}}$ and functional equation are established explicitly in [Reference BumpBum84, Chapter 8] and [Reference ButtcaneBut18]. Langlands proved these results in great generality.

We must keep track of the degenerate terms in the Fourier expansion of $E_{\min }^{(3)}(g; \alpha )$ for our applications. We prefer to apply the more compact form of the expansion (Lemma 3.13), rather than the fully explicit form in [Reference LiLi14], because the former better detects:

1. (1) the equivariance of the Fourier–Whittaker periods under unipotent translations; and

2. (2) the annihilation of undesirable degenerate terms in the Fourier expansion; see Lemma 6.5.

These two features yield much simpler formulae and a clearer presentation.

Recall that the Weyl group $W_{3}$ of $\text {GL}(3)$ consists of the permutation matrices

(3.29)

The actions of $w\in W_{3}$ on $\mathfrak {a}_{\mathbb {C}}$ are given by:

(3.30) $$ \begin{align} (\alpha_{1}, \alpha_{2}, \alpha_{3})^{I} &= (\alpha_{1}, \alpha_{2}, \alpha_{3}), \quad (\alpha_{1}, \alpha_{2}, \alpha_{3})^{w_{2}} = (\alpha_{2}, \alpha_{1}, \alpha_{3}), \quad (\alpha_{1}, \alpha_{2}, \alpha_{3})^{w_{3}} = (\alpha_{1}, \alpha_{3}, \alpha_{2}), \nonumber \\ (\alpha_{1}, \alpha_{2}, \alpha_{3})^{w_{4}} &= (\alpha_{3}, \alpha_{1}, \alpha_{2}), \quad (\alpha_{1}, \alpha_{2}, \alpha_{3})^{w_{5}} = (\alpha_{2}, \alpha_{3}, \alpha_{1}), \quad (\alpha_{1}, \alpha_{2}, \alpha_{3})^{w_{\ell}} = (\alpha_{3}, \alpha_{2}, \alpha_{1}). \end{align} $$

Translating [Reference BumpBum84, eqs. (3.10)–(3.15), (3.40) & (3.45)] into the notations of this article, we have

Definition 3.17. The degenerate Whittaker functions of $\overline {\mathrm {G}}_{3}$ are defined as follows:

(3.31) $$ \begin{align} W_{\alpha,\, w}^{(0,1)}(\mathbf{y}) \ &:= \ (y_{0}^{2}y_{1})^{(1-\alpha_{3}^{w})/2} \, \Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\, W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}(y_{1}) \end{align} $$

and

(3.32) $$ \begin{align} W_{\alpha,\, w}^{(1,0)}(\mathbf{y}) \ := \ (y_{0} y_{1}^2)^{(1+\alpha_{1}^{w})/2} \, \Gamma_{\mathbb{R}}\left( 1+\alpha_{1}^{w}-\alpha_{2}^{w}\right) \Gamma_{\mathbb{R}}\left(1+ \alpha_{1}^{w}-\alpha_{3}^{w}\right) W_{(\alpha_{2}^{w}-\alpha_{3}^{w})/2}(y_{0}), \end{align} $$

where $w\in W_{3}$ , $\alpha \in \mathfrak {a}_{\mathbb {C}}$ , $\mathbf {y}\in \mathrm {Y}^+$ , and $ W_{(\alpha _{i}^{w}-\alpha _{i+1}^{w})/2}$ ( $i=1,2$ ) are the standard Whittaker functions of $\overline {\mathrm {G}}_{2}$ .

It is clear that (3.31) and (3.32) grow polynomially as $y_{0}\to \infty $ and $y_{1}\to \infty $ , respectively. It is more convenient to consider the completed minimal parabolic Eisenstein series of $\Gamma _{3}$ , which is defined by

(3.33) $$ \begin{align} (E_{\min}^{(3)})^{*}(*; \; \ \alpha) \ := \ \prod_{1\le i<j\le 3} \, \Lambda(1+ \alpha_{i}-\alpha_{j})E_{\min}^{(3)}(*; \; \ \alpha). \end{align} $$

The completion factors in (3.33) simplify many formulae and ensure that the Hecke combinatorics of the non-degenerate Fourier coefficients coincide in both the cuspidal and Eisenstein cases. They also obviate the need to consider the zero-free region of the $\zeta $ -function in our argument. (Note: the formulae stated in [Reference LiLi14] and [Reference ButtcaneBut18] apply to the incomplete Eisenstein series.)

From this point onward, the automorphic form $\boldsymbol {\Phi }$ refers solely to (3.33). We record two results regarding the explicit evaluations of the degenerate Fourier–Whittaker periods of $\Phi $ .

Lemma 3.18. For any $\mathbf {y}\in \mathrm {Y}^{+}$ , $\alpha \in \mathfrak {a}_{\mathbb {C}}$ , and $n_{1}\in \mathbb {Z}- \{0\}$ , we have

(3.34) $$ \begin{align} \mathcal{W}_{(0,1)}(\mathbf{y}; \, \Phi) = \sum_{w=w_{2}, w_{4}, w_{\ell}} \ \zeta(w, \alpha, \psi_{(0,1)}) W^{(0,1)}_{\alpha,\, w}(\mathbf{y}) \end{align} $$

and

(3.35) $$ \begin{align} \mathcal{W}_{(n_{1},0)}(\mathbf{y}; \, \Phi) = \sum_{w=w_{3}, w_{5}, w_{\ell}} \ \frac{\zeta(w, \alpha, \psi_{(n_{1},0)})}{|n_{1}|^{1-\alpha_{3}^{w}}}\ W^{(1,0)}_{\alpha, \, w}(n_{1}y_{0},\, y_{1}), \end{align} $$

where

(3.36) $$ \begin{align} \zeta\big(w, \alpha, \psi_{(0,1)}\big) \ := \ \zeta\big( 1+\alpha_{2}^{w}-\alpha_{3}^{w}\big) \zeta\big( 1+\alpha_{1}^{w}-\alpha_{3}^{w}\big), \end{align} $$

and

(3.37) $$ \begin{align} \zeta\big(w, \alpha, \psi_{(n_{1},0)}\big) \ := \ \zeta\big(1+ \alpha_{1}^{w}-\alpha_{2}^{w}\big)\zeta\big(1+ \alpha_{1}^{w}-\alpha_{3}^{w}\big) \sigma_{\alpha_{2}^{w}-\alpha_{3}^{w}}(|n_{1}|). \end{align} $$

Proof. See [Reference BumpBum84, eqs. (7.7)–(7.8)] or [Reference ButtcaneBut18, eqs. (4.6)–(4.7)].

3.3.4 Automorphic L-functions

Suppose $\Phi $ is given by (3.33), and $\phi $ is an automorphic form of $\Gamma _{2}$ as in Section 3.3.1. Recall that

Definition 3.19. For $\sigma> 3/2$ , the standard L-functions of $\phi $ and $\Phi $ are defined, respectively, by

(3.38) $$ \begin{align} L(s, \phi)\ := \ \sum_{n=1}^{\infty} \ \frac{\mathcal{B}_{\phi}(n)}{n^{s}} \hspace{15pt} \text{and} \hspace{15pt} L(s, \Phi)\ := \ \sum_{n=1}^{\infty} \ \frac{\mathcal{B}_{\Phi}(1,n)}{n^{s}}. \end{align} $$

By [Reference GoldfeldGol06, Theorem 10.8.6], we have

(3.39) $$ \begin{align} L(s, \Phi) = \zeta(s+\alpha_{1}) \zeta(s+\alpha_{2})\zeta(s+\alpha_{3}). \end{align} $$

When $\phi $ is an even cusp form, we have the functional equation (see [Reference GoldfeldGol06, Proposition 3.13.5]):

(3.40) $$ \begin{align} \Lambda(s, \phi) = \prod_{\pm} \, \Gamma_{\mathbb{R}}(s\pm \mu) L(s, \phi) = \Lambda(1-s, \phi). \end{align} $$

Definition 3.20. For $\sigma> 3/2$ , the Rankin–Selberg L-function of $\Phi $ and $\phi $ is defined by

(3.41) $$ \begin{align} L(s, \phi\otimes \Phi) \ &:= \ \sum_{m_{1}=1}^{\infty} \ \sum_{m_{2}= 1}^{\infty} \frac{\mathcal{B}_{\phi}(m_{2}) \mathcal{B}_{\Phi}(m_{1}, m_{2} )}{ (m_{1}^2 m_{2})^{s}}. \end{align} $$

Lemma 3.21. Suppose $\phi $ is an even form of $\Gamma _{2}$ . Then for $\sigma> 3/2$ , we have

(3.42) $$ \begin{align} \mathcal{I}\big(s; \, \phi, \, \widetilde{\Phi}^{\mathrm{ND}}\big) = \frac{1}{2} \, \Lambda(s,\, \phi\otimes \widetilde{\Phi}), \end{align} $$

where

(3.43) $$ \begin{align} \Lambda(s,\, \phi\otimes \widetilde{\Phi}) \ &:= \ L_{\infty}(s,\, \phi\otimes \widetilde{\Phi})\cdot L(s, \,\phi\otimes \widetilde{\Phi}) \, := \, \prod_{\pm} \, \prod_{k=1}^{3} \, \Gamma_{\mathbb{R}}(s\pm \mu- \alpha_{k}) \cdot L(s,\, \phi\otimes \widetilde{\Phi}). \end{align} $$

Proof. Recall (3.23) and (2.11). Since $\widetilde {\Phi }^{\mathrm {ND}}$ is left-invariant by for any $\gamma \in \Gamma _{2}$ , the pairing in (3.42) is well-defined. The rest of the proof follows [Reference KwanKwa24, Proposition 5.15] (or [Reference GoldfeldGol06, Chapter 12.2]).

The following is an elementary consequence of Lemma 3.21:

Corollary 3.22. For $\sigma> 3/2$ , we have

(3.44) $$ \begin{align} \mathcal{I}\big(s; \, \phi,\, \widetilde{\Phi}^{\mathrm{ND}}\big) = \frac{1}{2} \ \prod_{i=1}^{3} \, \Lambda(s- \alpha_{i}, \phi), \end{align} $$

and

(3.45) $$ \begin{align} \mathcal{I}\big(s; \, E(*; \mu),\, \widetilde{\Phi}^{\mathrm{ND}}\big) = \frac{1}{2} \ \frac{ \prod_{\pm} \prod_{i=1}^{3} \, \Lambda( s\pm \mu -\alpha_{i} )}{\Lambda(1+2\mu)}. \end{align} $$

Proof. The standard L-functions $L(s, \phi )$ and $L(s, \Phi )$ admit Euler products of the form

$$ \begin{align*} L(s, \,\phi) = \prod_{p} \, \prod_{j=1}^{2} \, (1-\beta_{\phi, j}(p)p^{-s})^{-1} \hspace{15pt} \text{and} \hspace{15pt} L(s, \, \Phi) = \prod_{p} \, \prod_{k=1}^{3} \, (1- p^{-\alpha_{k}}p^{-s})^{-1} \end{align*} $$

for $\sigma> 3/2$ , and we have $\{ \beta _{\phi ,1}(p), \beta _{\phi ,2}(p) \} = \{ p^{\mu }, p^{-\mu } \}$ if $\phi = E^{*}(*;\mu )$ . The results follow from Cauchy’s identity in the form [Reference GoldfeldGol06, Proposition 7.4.12], i.e.,

$$ \begin{align*} L(s,\, \phi \otimes \Phi) = \prod_{p} \ \prod_{j=1}^{2} \ \prod_{k=1}^{3} \,\left(1- \beta_{\phi, j}(p)p^{-\alpha_{k}}\,p^{-s}\right)^{-1}. \end{align*} $$

5.1 CFKRS for orthogonal symmetry

For the shifted cubic moment $ \mathfrak {M}_{-\alpha }^{(3)}(s; H)$ , the CFKRS conjecture predicts $2^3=8$ main terms in total, which can be classified as follows [Reference Conrey and FazzariCF23]. Firstly, the diagonal term, or the ‘ $0$ -swap’ term, is of the form:

(5.1) $$ \begin{align} \zeta(1-\alpha_{1}-\alpha_{2})\zeta(1-\alpha_{1}-\alpha_{3})\zeta(1-\alpha_{2}-\alpha_{3}) \times \text{(archimedean part)}. \end{align} $$

Next, we alter the sign of exactly one of the components of $(-\alpha _{1}, -\alpha _{2}, -\alpha _{3})$ and we obtain the following three triples: $(\alpha _{1}, -\alpha _{2}, -\alpha _{3})$ , $(-\alpha _{1}, \alpha _{2}, -\alpha _{3})$ , $(-\alpha _{1}, -\alpha _{2}, \alpha _{3})$ . Accordingly, the following three main terms can be written down, up to the associated archimedean factors:

$$ \begin{align*} & \zeta(1+\alpha_{1}-\alpha_{2})\zeta(1+\alpha_{1}-\alpha_{3})\zeta(1-\alpha_{2}-\alpha_{3}),\\ &\zeta(1-\alpha_{1}+\alpha_{2})\zeta(1-\alpha_{1}-\alpha_{3})\zeta(1+\alpha_{2}-\alpha_{3}) ,\\ &\zeta(1-\alpha_{1}-\alpha_{2})\zeta(1-\alpha_{1}+\alpha_{3})\zeta(1-\alpha_{2}+\alpha_{3}). \end{align*} $$

These three terms are known as the ‘ $1$ -swap’ terms. The three ‘ $2$ -swap’ terms and the one ‘ $3$ -swap’ term can be written down similarly.

Proposition 3.14 pinpoints the sources of the main terms in a simple and unified way:

Observe that the ‘ $4=3+1$ ’ decomposition adopted in this work fits more naturally with the structures of the main terms for $ \mathfrak {M}_{-\alpha }^{(3)}(s; H)$ than the conventional ‘ $4=2+2$ ’ counterpart (see [Reference IvićIvi02, Reference FrolenkovFro20, Reference MotohashiMot93, Reference MotohashiMot97], for instance).

A salient feature of many explicit archimedean computations is the complete reduction of the Mellin–Barnes integrals to ratios of products of $\Gamma $ -factors, provided the arguments lie in certain ‘nice’ configurations. This phenomenon typically reflects deeper structures of the integrals. In our case, the reductions of the integral transforms $(\mathcal {F}_{\alpha }H)(s_{0}, s)$ and $J_{w, \alpha }(s; h)$ correspond exactly to the CFKRS conjecture for $ \mathfrak {M}_{-\alpha }^{(3)}(s; H)$ . This is the content of Theorem 1.4.

5.2 CFKRS for unitary symmetry

The CFKRS conjecture for the fourth moment of the Riemann $\zeta $ -function is stated in terms of the following Dirichlet series:

(5.2) $$ \begin{align} \mathcal{Z}_{A; B}(s) \ := \ \sum_{\substack{m,n \ge 1 \\m=n }} \ \frac{\tau_{A}(m)\tau_{B}(n)}{(mn)^{\frac{1}{2}+s}}, \end{align} $$

where $\tau _{A}(m):= \sum _{d_{1} d_{2}=m} \ d_{1}^{-\alpha _{1}} d_{2}^{-\alpha _{2}}$ if $A= \{ \alpha _{1}, \alpha _{2} \}$ , and similarly for $\tau _{B}(n)$ if $B= \{\beta _{1}, \, \beta _{2}\}$ . The series (5.2) comes up naturally, as the recipe of [Reference Conrey, Farmer, Keating, Rubinstein and SnaithCFK+05, Sections 1.7 and 2] captures only the diagonal. For $\sigma>0$ , we have

(5.3) $$ \begin{align} \mathcal{Z}_{A;B}(s) \ \ := \ \ \prod_{\alpha \in A} \ \prod_{\beta \in B} \ \zeta(1+2s+\alpha+\beta)\cdot \widetilde{\mathcal{A}}_{AB}(s), \end{align} $$

and the infinite Euler product $ \widetilde {\mathcal {A}}_{AB}(s)$ converges absolutely on $\sigma> -\delta $ for some small $\delta>0$ (see [Reference Conrey, Farmer, Keating, Rubinstein and SnaithCFK+05, Theorem 2.4.2.1]). The full set of main terms for the shifted moment

(5.4) $$ \begin{align} \int_{\mathbb{R}} \ \eta(t)\prod_{\alpha\in A} \ \zeta\left(\tfrac12 +it + \alpha\right) \ \prod_{\beta\in B} \zeta\left(\tfrac12 -it+ \beta\right) \ dt \end{align} $$

is conjectured to be

(5.5) $$ \begin{align} \int_{\mathbb{R}} \, \eta(t) \, \Big\{ \sum_{\ell =0}^{2} \ \sum_{\substack{U\subset A, \ V\subset B\\ |U|=|V|=\ell}} \ \left(\frac{t}{2\pi}\right)^{-U-V} \mathcal{Z}_{A_{U}; \,B_{V} }(0) \Big\} \ dt, \end{align} $$

where $A_{U}:= A\setminus U\cup V^{-}$ , $B_{V}:= B\setminus V \cup U^{-}$ , $U^{-}:= \{-u : u \in U\}$ (similarly for $V^{-}$ ), and $-U-V:= - \sum _{\alpha \in U} \alpha - \sum _{\beta \in V} \beta $ . The expression inside the braces in (5.5) can be written out more explicitly as follows:

(5.6) $$ \begin{align} & Z_{\{\alpha_{1}, \alpha_{2} \}; \left\{ \beta_{1}, \beta_{2}\right\}}(0) + Z_{\{-\beta_{1}, \alpha_{2} \}; \left\{ -\alpha_{1}, \beta_{2}\right\}}(0) \left(\frac{t}{2\pi}\right)^{-\alpha_{1}-\beta_{1}} \nonumber\\ &\quad + Z_{\{-\beta_{2}, \alpha_{2} \}; \left\{ \beta_{1}, -\alpha_{1}\right\}}(0) \left(\frac{t}{2\pi}\right)^{-\alpha_{1}-\beta_{2}} + Z_{\{\alpha_{1}, -\beta_{1} \}; \left\{ -\alpha_{2}, \beta_{2}\right\}}(0) \left(\frac{t}{2\pi}\right)^{-\alpha_{2}-\beta_{1}} \nonumber\\ &\quad + Z_{\{\alpha_{1}, -\beta_{2} \}; \left\{ \beta_{1}, -\alpha_{2}\right\}}(0) \left(\frac{t}{2\pi}\right)^{-\alpha_{2}-\beta_{2}} + Z_{\{-\beta_{1}, -\beta_{2} \}; \left\{ -\alpha_{1}, -\alpha_{2}\right\}}(0) \left(\frac{t}{2\pi}\right)^{-\alpha_{1}-\alpha_{2}-\beta_{1}-\beta_{2}}. \end{align} $$

The first and last terms of (5.6) are known as the ‘ $0$ -swap’ and ‘ $2$ -swap’ terms, respectively, whereas the middle four terms of (5.6) are the ‘ $1$ -swap’ terms. These six terms are the ‘non-oscillatory’ terms that remain in the recipe, out of the total $2^4=16$ terms arising from all possible sign combinations of the shifts. Furthermore, the Ramanujan identity provides a nice formula when $|A|=|B|=2$ :

(5.7) $$ \begin{align} \widetilde{\mathcal{A}}_{AB}(s) \, = \, \zeta\Big( 2+ 2s+ \sum_{\alpha\in A} \ \alpha + \sum_{\beta\in B} \ \beta\Big)^{-1}. \end{align} $$

6.1 Spectral expansion and Eisenstein contribution

Definition 6.1. Let $h\in C^{\infty }(0,\infty )$ and $\psi (x):= e(x)$ , which are regarded as functions on $\overline {\mathrm {G}}_{2}$ as follows:

The associated Poincaré series of $\Gamma _{2}$ is defined by

(6.1) $$ \begin{align} P(g;\, h) \ := \ \sum_{\gamma\in [\Gamma_{2}]} (h\psi)(\gamma g) \hspace{20pt} (g \, \in \, \mathrm{G}_{2}), \end{align} $$

provided it converges absolutely.

If the bound $h(y) \,\ll \, y^{1+\epsilon }(1+y)^{-1/2-2\epsilon }$ is satisfied for any $y>0$ , then the Poincaré series $P(g;\, h)$ converges absolutely and uniformly on every Siegel set, and is an $L^2$ -function. Recall that we take $h:= H^{\flat }$ with $H\in \mathcal {C}_{\eta }$ and $\eta>40$ in this article; see (3.13) and Assumption 1.2. It follows from (3.14) that the desired bound for h holds.

Lemma 6.2. Let $ \mathfrak {M}_{-\alpha }^{(3)}(s; H)$ be defined in (1.6). For $\sigma> 3/2$ and $H\in \mathcal {C}_{\eta }$ , we have

(6.2) $$ \begin{align} 2 \cdot \mathcal{I}\big(s; \, P(\,*\,; h), \, \widetilde{\Phi}^{\mathrm{ND}}\big) = \mathfrak{M}_{-\alpha}^{(3)}(s; H). \end{align} $$

The cuspidal part of $ \mathfrak {M}_{-\alpha }^{(3)}(s; H)$ evidently admits an entire continuation. The Eisenstein part, initially defined on $\sigma> 1+\epsilon _{0}/2$ , also admits a continuation, though this is less immediate. Define

(6.3) $$ \begin{align} \mathcal{C}_{\alpha}(s; \, \kappa; \, H) \ \ := \ \ \int_{(\kappa)} \ H(\mu) \, \frac{ \prod_{i=1}^{3} \, \prod_{\pm} \ \Lambda( s \pm \mu -\alpha_{i} )}{\Lambda(1+2\mu) \Lambda(1-2\mu)} \, \frac{d\mu}{2\pi i }, \end{align} $$

and let $ \mathcal {R}_{\alpha }(s; H)$ be as in (1.13). Since $H\in \mathcal {C}_{\eta }$ , the integrand in (6.3) is holomorphic on $|\operatorname {\mathrm {Re}} \mu |<2\eta $ , except for the poles at

(6.4) $$ \begin{align} \mu \in \big\{-s + \alpha_{i}, \ 1 - s + \alpha_{i}\big\}_{i=1}^{3} \hspace{20pt} \text{ and } \hspace{20pt} \mu \ \in \ \big\{ s - \alpha_{i}, \ s-1 -\alpha_{i}\big\}_{i=1}^{3}, \end{align} $$

as well as the zeros of $\Lambda (1\pm 2\mu )$ .

Proof. This is an adaptation of [Reference MotohashiMot97, p. 118] and we sketch the argument for our setting. Firstly, suppose $1+\epsilon _{0}/2< \sigma < 1+\epsilon _{0}$ . Since $H\in \mathcal {C}_{\eta }$ , we may shift the line of integration to $\operatorname {\mathrm {Re}} \mu =\eta> 40$ , picking up the residues at the second set of poles in (6.4) and those at the poles of $\Lambda (1-2\mu )^{-1}$ . We have

(6.5) $$ \begin{align} \mathcal{C}_{\alpha}(s; \, 0; \, H) \, = \, - \sum_{i=1}^{3} \, (\operatorname{\mathrm{Res}}_{\mu=s-1-\alpha_{i}} \, + \, \operatorname{\mathrm{Res}}_{\mu=s-\alpha_{i}}) \, - \, \sum_{\rho} \ \operatorname{\mathrm{Res}}_{\mu= \frac{1-\rho}{2}} \ + \ \mathcal{C}_{\alpha}(s; \, \eta; \, H), \end{align} $$

where $\rho $ runs over all nontrivial zeros of $\zeta (s)$ .

Secondly, observe that $\mathcal {C}_{\alpha }(s; \, \eta; \, H)$ is holomorphic on $1-\eta + \epsilon _{0} < \sigma < 1+ \epsilon _{0}$ , using the holomorphy of $\Lambda (s)$ on $\{\sigma <0\} \cup \{\sigma>1\}$ , and the fact that $\Lambda (1\pm 2\mu )\neq 0$ on $\operatorname {\mathrm {Re}} \mu =\eta $ . Together with the residual terms of (6.5), a meromorphic continuation of $(\mathcal {C}_{\alpha }H)(s)$ to $\sigma> 1-\eta +\epsilon _{0}$ is obtained.

Thirdly, we restrict the continuation obtained to the smaller region $\epsilon _{0} < \sigma < 1-\epsilon _{0}$ . The poles $\mu =1-s+\alpha _{i}$ , $\mu = s-\alpha _{i}$ , and the zeros of $\Lambda (1-2\mu )$ lie between the lines $\operatorname {\mathrm {Re}} \mu =0$ and $\operatorname {\mathrm {Re}} \mu = \eta $ . We shift the line of integration of $\mathcal {C}_{\alpha }(s; \, \eta; \, H)$ back to $\operatorname {\mathrm {Re}} \mu =0$ . The residual contributions of $\mu =s-\alpha _{i}$ and $\mu =(1-\rho )/2$ in (6.5) are now cancelled, leaving

(6.6) $$ \begin{align} - \sum_{i=1}^{3} \, \operatorname{\mathrm{Res}}_{\mu=s-1-\alpha_{i}} \, + \, \sum_{i=1}^{3} \, \operatorname{\mathrm{Res}}_{\mu=1-s+\alpha_{i}} \, + \, \mathcal{C}_{\alpha}(s; \, 0; \, H) \end{align} $$

as the meromorphic continuation of $\mathcal {C}_{\alpha }(s; 0; H)$ to the domain $\epsilon _{0} < \sigma < 1-\epsilon _{0}$ . A little calculation shows that $\mathcal {R}_{\alpha }(1-s; H) = - \sum _{i=1}^{3} \, \operatorname {\mathrm {Res}}_{\mu =s-1-\alpha _{i}} \, + \, \sum _{i=1}^{3} \, \operatorname {\mathrm {Res}}_{\mu =1-s+\alpha _{i}}$ , and this completes the sketch.

The following is immediate from Lemmas 6.2 and 6.3.

6.2 Unfolding and rearrangement

By unfolding, we have the equality

(6.7) $$ \begin{align} \mathcal{I}\big(s; \, P(\,*\,; h), \, \widetilde{\Phi}^{\mathrm{ND}}\big) = \int_{0}^{\infty} \int_{0}^{\infty} \, h(y_{1}) (y_{0}^2y_{1})^{s-\frac{1}{2}} \mathcal{P}_{\psi^{-1}}\big(\mathbf{y}; \, \widetilde{\Phi}^{\mathrm{ND}}\big) \ d^{*}\mathbf{y}, \end{align} $$

where the right-hand side converges absolutely on $1+\epsilon _{0}<\sigma < 4$ ; see [Reference KwanKwa24, Section 6B]. In what follows, we perform our analysis on suitable vertical strips within the half-plane $\sigma <4$ .

To avoid confusion with computations in the literature (e.g., regarding the sign convention of parameters and normalization of Fourier coefficients), it is better to consider instead

(6.8) $$ \begin{align} \mathcal{J}(s; \, h, \, \Phi) \ := \ \int_{0}^{\infty} \int_{0}^{\infty} \ h(y_{1}) (y_{0}^2y_{1})^{s-\frac{1}{2}} \mathcal{P}_{\psi}(\mathbf{y}; \, \Phi^{\mathrm{ND}}) \ d^{*}\mathbf{y} \end{align} $$

in the main calculations of Sections 6.3–6.4. As is apparent from [Reference KwanKwa24, (6.5), (6.7), (6.8)], we have

(6.9) $$ \begin{align} \mathcal{I}\big(s; \, P(\,*\,; h), \, \widetilde{\Phi}^{\mathrm{ND}}\big) \, = \, \mathcal{J}(s; \, h, \, \widetilde{\Phi}). \end{align} $$

The next lemma explains why our method does not require any elaborate regularization of periods.

Lemma 6.5. For any $\mathbf {y}\in \mathrm {Y}^{+}$ , we have

(6.10) $$ \begin{align} \mathcal{W}_{(0,1)}(\mathbf{y}; \, \Phi)\ - \ \sum_{n_{2}\neq 0} \ \mathcal{P}_{\psi}\big(\mathbf{y}; \, \mathcal{W}_{(0, n_{2})}(\,*\,; \, \Phi) \big) \, = \, 0. \end{align} $$

Proof. Since $\mathcal {W}_{(0, n_{2})}(\mathbf {n} \mathbf {y}; \, \Phi ) = \psi _{n_{2}}(\mathbf {n}) \mathcal {W}_{(0, n_{2})}(\mathbf {y};\, \Phi )$ for $\mathbf {n}\in \mathrm {N}_{12}$ and $\mathbf {y}\in \mathrm {Y}^{+}$ , the result follows from

$$ \begin{align*} \sum_{n_{2}\neq 0} \, \mathcal{P}_{\psi}\big(\mathbf{y}; \,\mathcal{W}_{(0, n_{2})}(*; \, \Phi) \big) = \sum_{n_{2}\neq 0} \, \mathcal{W}_{(0, n_{2})}(\mathbf{y};\, \Phi) \, \int_{0}^{1} e\left( (n_{2}-1) u\right) \, du = \mathcal{W}_{(0, 1)}(\mathbf{y};\, \Phi). \end{align*} $$

We have the following decompositions of our periods:

Proposition 6.6. For any $\mathbf {y}\in \mathrm {Y}^{+}$ , we have

(6.11) $$ \begin{align} \mathcal{P}_{\psi}\big(\mathbf{y}; \, \Phi^{\mathrm{ND}}\big) = \mathcal{P}^{\mathrm{reg}}\big(\mathbf{y}; \, \Phi\big) + \mathcal{P}_{(0,1)}^{\min}(\mathbf{y}; \, \Phi) - \mathcal{P}_{(*,0)}^{\min}(\mathbf{y}; \, \Phi), \end{align} $$

and

(6.12) $$ \begin{align} \mathcal{J}(s; \, h, \, \Phi) \, = \, \mathcal{J}^{\mathrm{reg}}(s; \, h, \, \Phi) \, + \, \mathcal{J}_{(0,1)}^{\min}(s; \, h, \, \Phi) \, - \, \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \Phi), \end{align} $$

where

(6.13) $$ \begin{align} \mathcal{P}^{\mathrm{reg}}\big(\mathbf{y}; \, \Phi\big) \, &:= \, \sum_{a_{1}\in \mathbb{Z}-\{0\}} \, \sum_{a_{0}\in \mathbb{Z}} \ \mathcal{W}_{(a_{1},1)}\big( \mathbf{n}_{32}(-a_{0}) \mathbf{y}; \, \Phi\big), \end{align} $$

(6.14) $$ \begin{align}\ \mathcal{P}_{(0,1)}^{\min}(\mathbf{y}; \, \Phi) \ &:= \ \sum_{a_{0}\in \mathbb{Z}-\{0\}} \ \mathcal{W}_{(0,1)}\left( \mathbf{n}_{32}(-a_{0}) \mathbf{y}; \, \Phi\right), \end{align} $$

(6.15) $$ \begin{align} \mathcal{P}_{(*,0)}^{\min}(\mathbf{y}; \, \Phi) \, &:= \, \sum_{\gamma\in [\Gamma_{2}]} \, \sum_{n_{1}= 1}^{\infty} \, \mathcal{P}_{\psi}\big(\mathbf{y}; \, \gamma\cdot \mathcal{W}_{(n_{1},0)}(*; \, \Phi) \big), \end{align} $$

and $(\gamma \cdot f)(g):= f(\gamma g)$ , and for $\dagger \in \{\mathrm {reg}, \, \min \}$ and $\bullet \in \{ \quad ,\, (0,1),\, (*,0)\}$ , we set

(6.16) $$ \begin{align} \mathcal{J}^{\dagger}_{\bullet}(s; \, h, \, \Phi) \, := \, \int_{0}^{\infty} \int_{0}^{\infty} \, h(y_{1}) (y_{0}^2y_{1})^{s-\frac{1}{2}} \mathcal{P}^{\dagger}_{\bullet}\big(\mathbf{y}; \, \Phi\big) \ d^{*}\mathbf{y}. \end{align} $$

6.3 Degenerate terms – I

In this section, we evaluate $\mathcal {J}_{(0,1)}^{\min }(s; \, h, \, \Phi ) $ defined in Proposition 6.6. The computation produces three terms, which will be identified in Section 6.8 with terms appearing in the CFKRS conjecture for the cubic moment.

Proposition 6.7. Suppose $1/2+\epsilon _{0}< \sigma < 4$ . Then $ \mathcal {J}_{(0,1)}^{\min }(s; h, \Phi ) $ is equal to

(6.17) $$ \begin{align} 2 \cdot \Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right)\sum_{w=w_{2}, w_{4}, w_{\ell}} \ \zeta(w, \alpha, \psi_{(0,1)}) \zeta(2s+\alpha_{1}^{w}+\alpha_{2}^{w}) J_{w, \alpha}(s; h), \end{align} $$

where

(6.18) $$ \begin{align} J_{w, \alpha}(s; h) := \int_{0}^{\infty} \int_{0}^{\infty} h(y_{1}) y_{1}^{s-\frac{\alpha_{3}^{w}}{2}} y_{0}^{2s-\alpha_{3}^{w}} (1+y_{0}^2)^{-\frac{3}{4}(1-\alpha_{3}^{w})} W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}\Big(y_{1}\sqrt{1+y_{0}^2}\Big) \, d^{*}\mathbf{y}. \end{align} $$

Proof. We begin with the matrix identity

(6.19) $$ \begin{align} \mathbf{n}_{32}(-a_{0}) \mathbf{y} \ \equiv \ \mathbf{n}_{23}\Big(-\frac{a_{0}y_{0}^2}{1+(a_{0}y_{0})^2}\Big) \mathbf{y}\Big(\frac{y_{0}}{1+(a_{0}y_{0})^2}, \, y_{1} \sqrt{1+(a_{0}y_{0})^2}\Big) \hspace{15pt} (\bmod\, \mathrm{K}_{3}\cdot \mathbb{R}_{>0}^{\times}). \end{align} $$

Applying this with (3.31), (3.34) and (6.14), we have

(6.20) $$ \begin{align} \mathcal{J}_{(0,1)}^{\min}(s; \, h, \, \Phi) &= \Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right) \Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right) \sum_{w=w_{2}, w_{4}, w_{\ell}} \ \zeta(w, \alpha, \psi_{(0,1)}) \nonumber\\ & \cdot \ \sum_{a_{0}\neq 0} \ \int_{0}^{\infty} \int_{0}^{\infty} \ h(y_{1})(y_{0}^2y_{1})^{s-\frac{1}{2}} \Big(\frac{y_{0}}{1+(a_{0}y_{0})^2}\Big)^{1-\alpha_{3}^{w}} \Big(y_{1}\sqrt{1+(a_{0}y_{0})^2}\Big)^{\frac{1-\alpha_{3}^{w}}{2}} \nonumber\\ &\cdot \, W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}\Big(y_{1}\sqrt{1+(a_{0}y_{0})^2}\Big) \, d^{*}\mathbf{y}. \end{align} $$

Making a change of variables $y_{0}\to |a_{0}|^{-1} y_{0}$ , we find that $ \mathcal {J}_{(0,1)}^{\min }(s; \, h, \, \Phi ) $ is equal to

$$ \begin{align*} 2\Gamma_{\mathbb{R}}\left(1+\alpha_{2}^{w}-\alpha_{3}^{w}\right)\Gamma_{\mathbb{R}}\left(1+\alpha_{1}^{w}-\alpha_{3}^{w}\right) \sum_{w=w_{2}, w_{4}, w_{\ell}} \ \zeta(w, \alpha, \psi_{(0,1)}) \zeta(2s+\alpha_{1}^{w}+\alpha_{2}^{w}) J_{w, \alpha}(s; h) \end{align*} $$

for $ 1/2+\epsilon _{0}<\sigma < 4$ . This completes the proof.

Proof. Recall that $H:= h^{\#} \in \mathcal {C}_{\eta }$ and $\eta> 40$ . Using the bound

$$ \begin{align*} W_{(\alpha_{1}^{w}-\alpha_{2}^{w})/2}(y) \, \ll_{A} \, y^{-A} \end{align*} $$

with $A> 2\sigma -3/2+\epsilon _{0}$ and $\eta -1 -\epsilon _{0}> |A-\sigma | $ , observe that $J_{w, \alpha }(s; h)$ converges absolutely whenever $\epsilon _{0}< \sigma < 4$ . With the analytic continuation of the $\zeta $ -function, the right-hand side of (6.17) serves as a continuation of $ \mathcal {J}_{(0,1)}^{\min }(s; \, h, \, \Phi ) $ to $\epsilon _{0}< \sigma < 4$ . It is clearly holomorphic on $\epsilon _{0}< \sigma < 4$ , except for the simple poles coming from the factor $\zeta (2s+\alpha _{1}^{w}+\alpha _{2}^{w})$ ( $w\in \{w_{2}, w_{4}, w_{\ell }\}$ ). This completes the proof.

Proposition 6.9. On the region $\epsilon _{0} < \sigma < 1-\epsilon _{0}$ , we have

(6.21) $$ \begin{align} J_{w, \alpha}(s; h) & = \frac{1}{4} \Gamma_{\mathbb{R}}(1+\alpha_{2}^{w}-\alpha_{3}^{w})^{-1}\Gamma_{\mathbb{R}}(1+\alpha_{1}^{w}-\alpha_{3}^{w})^{-1} \nonumber\\ &\quad \cdot \ \int_{(0)} \, H(\mu)\prod_{\pm} \, \Gamma_{\mathbb{R}}(1-s\pm \mu-\alpha_{3}^{w}) \prod_{i=1}^{2}\, \Gamma_{\mathbb{R}}(s\pm \mu+\alpha_{i}^{w}) \, d^{W}\mu. \end{align} $$

Proof. We begin with the following initial domain:

(6.22) $$ \begin{align} \hspace{20pt} 1/2+\epsilon_{0} \ < \ \sigma \ < \ 4, \ \hspace{15pt} \sigma_{u} \ > \ 2\sigma-3/2+ \epsilon_{0}, \hspace{15pt} 40-\epsilon_{0} \ > \ |\sigma_{u}-\sigma+1|. \end{align} $$

It follows that

$$ \begin{align*} J_{w, \alpha}(s; h) & = \frac{1}{2} \, \int_{(\sigma_{u})} \ \prod_{\pm} \, \Gamma_{\mathbb{R}}\Big(u+ \frac{1\pm (\alpha_{1}^{w}-\alpha_{2}^{w})}{2}\Big) \Big( \int_{0}^{\infty} \, h(y_{1}) y_{1}^{s-1-\frac{\alpha_{3}^{w}}{2}-u} \, d^{\times} y_{1}\Big) \\ &\quad \cdot \, \Big( \int_{0}^{\infty} \, y_{0}^{2s-\alpha_{3}^{w}} (1+y_{0}^2)^{-\frac{3}{4}(1-\alpha_{3}^{w})-\frac{u}{2}} \, d^{\times} y_{0}\Big) \ \ \frac{du}{2\pi i}.\end{align*} $$

Indeed, the first two conditions of (6.22) ensure that the $y_{0}$ -integral converges absolutely, and the Euler beta integral formula and (3.4) can be applied (with Mellin inversion). The last condition of (6.22) guarantees the absolute convergence of the $y_{1}$ -integral. As a result, we have

(6.23) $$ \begin{align} J_{w, \alpha}(s; h) &= \frac{1}{4} \Gamma\Big(s-\frac{\alpha_{3}^{w}}{2}\Big) \, \int_{(\sigma_{u})} \ \, \frac{\prod_{\pm} \, \Gamma_{\mathbb{R}}(u+ \frac{1\pm (\alpha_{1}^{w}-\alpha_{2}^{w})}{2})\Gamma(\frac{3-\alpha_{3}^{w}}{4}-s+ \frac{u}{2})}{\Gamma( \frac{3}{4}(1-\alpha_{3}^{w})+\frac{u}{2})} \nonumber \\& \quad\times\widetilde{h} \Big(s-1-\frac{\alpha_{3}^{w}}{2}-u\Big) \ \frac{du}{2\pi i}. \end{align} $$

Upon restricting to the strip $1/2+\epsilon _{0} < \sigma < 1-2\epsilon _{0}$ , we pick $\sigma _{u} \in (2\sigma -3/2 + \epsilon _{0}, \, \sigma - 1/2-\epsilon _{0})$ . The contour $\operatorname {\mathrm {Re}} u=\sigma _{u}$ satisfies the Barnes convention, and the condition for (3.16) is satisfied. We have

$$ \begin{align*} \widetilde{h}\Big(s-1-\frac{\alpha_{3}^{w}}{2}-u\Big) = \frac{1}{2}\, \int_{(0)} \ H(\mu)\, \prod_{\pm} \, \Gamma_{\mathbb{R}}\Big(s-\frac{1+\alpha_{3}^{w}}{2}-u\pm\mu\Big) \ d^{W}\mu, \nonumber \end{align*} $$

and hence, $J_{w, \alpha }(s; h) $ is equal to

$$ \begin{align*} & \frac{\pi^{-(s-\frac{\alpha_{3}^{w}}{2})}}{8} \ \Gamma\Big(s-\frac{\alpha_{3}^{w}}{2}\Big) \int_{(0)} \, H(\mu) \\ &\quad \times \int_{(\sigma_{u})} \ \frac{\Gamma\Big(\frac{3-\alpha_{3}^{w}}{4}-s+ \frac{u}{2}\Big)\prod_{\pm} \, \Gamma\Big(\frac{u}{2}+ \frac{1\pm (\alpha_{1}^{w}-\alpha_{2}^{w})}{4}\Big)\Gamma\Big(\frac{s-\frac{1+\alpha_{3}^{w}}{2}-u\pm \mu}{2}\Big) }{\Gamma( \frac{3}{4}(1-\alpha_{3}^{w})+\frac{u}{2})} \ \frac{du}{2\pi i} \, d^{W}\mu.\nonumber \end{align*} $$

We make a change of variables $u \to 2u$ , and take

$$ \begin{align*} & (a,b,c; d,e ) \\ &\quad := \Big( \frac{1+(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}, \ \frac{1-(\alpha_{1}^{w}-\alpha_{2}^{w})}{4}, \ \frac{3-\alpha_{3}^{w}}{4}-s; \ \frac{s+\mu}{2}-\frac{1+\alpha_{3}^{w}}{4}, \ \frac{s-\mu}{2}-\frac{1+\alpha_{3}^{w}}{4} \Big) \end{align*} $$

and verify that

$$ \begin{align*} (a+ b)+c + (d +e) = \frac{1}{2} + \Big(\frac{3-\alpha_{3}^{w}}{4} - s \Big) + \Big(s -\frac{1+\alpha_{3}^{w}}{2}\Big) = \frac{3}{4}\ (1-\alpha_{3}^{w}) (:=) f. \end{align*} $$

We apply Lemma 3.1 to the u-integral. Observe that a pair of factors $\Gamma (s-\alpha _{3}^{w}/2)$ cancels, and (6.21) follows from $\alpha _{1}^{w}+\alpha _{2}^{w}+\alpha _{3}^{w}=0$ and conversion with $\Gamma _{\mathbb {R}}(s):= \pi ^{-s/2}\Gamma (s/2)$ . The validity of (6.21) on the larger domain $\epsilon _{0}<\sigma <1-\epsilon _{0}$ follows from analytic continuation. This completes the proof.

Corollary 6.10. Let $\mathcal {M}_{-\alpha }^{1}(s; H)$ be defined by (1.10). We have

(6.24) $$ \begin{align} \mathcal{J}_{(0,1)}^{\min}(s; h, \widetilde{\Phi}) = \frac{1}{2}\,\mathcal{M}_{-\alpha}^{1}(s; H). \end{align} $$

6.4 Degenerate terms – II

The main task of this section is to evaluate $\mathcal {J}_{(*,0)}^{\min }(s; \, h, \, \Phi ) $ defined in Proposition 6.6. Interestingly, the $\gamma $ -sum in (6.15), which arises from the $\Gamma _{3}$ Fourier expansion of $\Phi $ , transforms into the $\Gamma _{2}$ Eisenstein series, and produces three of the main terms for the fourth moment of the $\zeta $ -function. This is also essential for the analytic continuation of (6.7). Such phenomena do not appear in the first two moments of $\mathrm {GL}(2)\ L$ -functions ([Reference MotohashiMot92]).

Proposition 6.11. For $1/2+\epsilon _{0}< \sigma < 4$ , we have

(6.25) $$ \begin{align} \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \widetilde{\Phi})\ = \ \mathcal{R}_{-\alpha}(s; H). \end{align} $$

Proof. We have

(6.26) $$ \begin{align} & \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \Phi) \nonumber \\ &\quad = \int_{0}^{\infty} \int_{0}^{\infty} \ h(y_{1})(y_{0}^2y_{1})^{s-\frac{1}{2}} \sum_{\gamma\in [\Gamma_{2}]} \ \sum_{n_{1}= 1}^{\infty} \, \int_{[\mathrm{N}_{12}]} \, \mathcal{W}_{(n_{1},0)}( \gamma\mathbf{n}_{12} \mathbf{y}; \, \Phi) \overline{\psi(\mathbf{n}_{12})} \ d\mathbf{n}_{12} \, d^{*}\mathbf{y}. \end{align} $$

For any $\gamma \in \Gamma _{2}$ , we have the Iwasawa decomposition:

$$ \begin{align*} \gamma \begin{pmatrix} 1 & u \\ & 1 \end{pmatrix} \begin{pmatrix} y_{0}y_{1} & \\ & y_{0} \end{pmatrix} \equiv y_{0}y_{1}^{1/2} \begin{pmatrix} 1 & \operatorname{\mathrm{Re}} \gamma z\\ & 1 \end{pmatrix} \begin{pmatrix} (\operatorname{\mathrm{Im}} \gamma z)^{1/2} & \\ & (\operatorname{\mathrm{Im}} \gamma z)^{-1/2} \end{pmatrix} \quad \left(\bmod\, \mathrm{SO}(2)\right).\end{align*} $$

where $z := u + i y_{1}$ . Using this together with the equivariance of $ \mathcal {W}_{(n_{1}, 0)}$ , observe that

$$ \begin{align*} \mathcal{W}_{(n_{1},0)}( \gamma\mathbf{n}_{12}(u) \mathbf{y}; \, \Phi) \, = \, \mathcal{W}_{(n_{1}, 0)}\big[\big(y_{0} (y_{1}/\operatorname{\mathrm{Im}} \gamma z)^{1/2} , \, \operatorname{\mathrm{Im}} \gamma z \big); \, \Phi\big]. \end{align*} $$

By this and Lemma 3.18, we have

$$ \begin{align*} \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \Phi) &= \sum_{w=w_{3}, w_{5}, w_{\ell}} \, \sum_{\gamma\in [\Gamma_{2}]} \, \sum_{n_{1}= 1}^{\infty} \, \frac{\zeta(w, \alpha, \psi_{(n_{1},0)})}{|n_{1}|^{1-\alpha_{3}^{w}}} \nonumber\\ & \quad\times \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{1} \, h(y_{1}) (y_{0}^2y_{1})^{s-\frac{1}{2}} W^{(1,0)}_{\alpha,\, w}\big(n_{1}y_{0} (y_{1}/\operatorname{\mathrm{Im}} \gamma z)^{1/2}, \, \operatorname{\mathrm{Im}} \gamma z\big) e(-u) \ du\, d^{*}\mathbf{y}. \end{align*} $$

It follows from the change of variables $y_{0} \to n_{1}^{-1} (y_{1}/ \operatorname {\mathrm {Im}} \gamma z)^{-1/2} y_{0}$ that

$$ \begin{align*} \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \Phi) &= \sum_{w=w_{3}, w_{5}, w_{\ell}} \Big(\sum_{n_{1}= 1}^{\infty} \ \frac{\zeta(w, \alpha, \psi_{(n_{1},0)})}{|n_{1}|^{1-\alpha_{3}^{w} +(2s-1)}} \Big) \nonumber\\ & \quad \cdot \sum_{\gamma\in [\Gamma_{2}]}\, \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{1} \ h(y_{1})(y_{0}^2 \operatorname{\mathrm{Im}} \gamma z)^{s-\frac{1}{2}} W^{(1,0)}_{\alpha,\, w}(y_{0}, \, \operatorname{\mathrm{Im}} \gamma z) e(-u) \ du\, d^{*}\mathbf{y}. \nonumber \end{align*} $$

(Note: z does not depend on $y_{0}$ !) Now, substituting (3.32) for $ W^{(1,0)}_{\alpha , \, w}(\cdots )$ , we find that

(6.27) $$ \begin{align} \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \Phi) & = \sum_{w=w_{3}, w_{5}, w_{\ell}} \Gamma_{\mathbb{R}}\left( 1+\alpha_{1}^{w}-\alpha_{2}^{w}\right) \Gamma_{\mathbb{R}}\left(1+ \alpha_{1}^{w}-\alpha_{3}^{w}\right) \ \nonumber\\ &\quad \cdot \Big(\sum_{n_{1}= 1}^{\infty} \frac{\zeta(w, \alpha, \psi_{(n_{1},0)})}{|n_{1}|^{2s-\alpha_{3}^{w} }} \Big) \Big( \int_{0}^{\infty} \ W_{(\alpha_{2}^{w}-\alpha_{3}^{w})/2}(y_{0}) y_{0}^{2s-1+ (1+\alpha_{1}^{w})/2} \ d^{\times} y_{0}\Big) \ \nonumber\\ & \quad \cdot \sum_{\gamma\in [\Gamma_{2}]} \ \int_{0}^{\infty} \int_{0}^{1} \ h(y_{1})(\operatorname{\mathrm{Im}} \gamma z)^{s-\frac{1}{2}+(1+\alpha_{1}^{w})} e(-u) \ du\, \frac{dy_{1}}{y_{1}^2}. \end{align} $$

In (6.27), we move the $\gamma $ -sum inside the double integral, and thus,Footnote ¹¹

(6.28) $$ \begin{align} \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \Phi) & = \sum_{w=w_{3}, w_{5}, w_{\ell}} \Gamma_{\mathbb{R}}\left( 1+\alpha_{1}^{w}-\alpha_{2}^{w}\right) \Gamma_{\mathbb{R}}\left(1+ \alpha_{1}^{w}-\alpha_{3}^{w}\right) \nonumber\\ & \quad \cdot \Big(\sum_{n_{1}= 1}^{\infty} \frac{\zeta(w, \alpha, \psi_{(n_{1},0)})}{|n_{1}|^{2s-\alpha_{3}^{w} }} \Big) \Big( \int_{0}^{\infty} \ W_{(\alpha_{2}^{w}-\alpha_{3}^{w})/2}(y_{0}) y_{0}^{2s-1+ (1+\alpha_{1}^{w})/2} \ d^{\times} y_{0}\Big)\nonumber\\ &\quad \cdot \Big(\, 2\ \int_{0}^{\infty} \, h(y_{1}) \, \int_{0}^{1} \, E\left(z; s+ \alpha_{1}^{w} \right)e(-u) \, du\, \frac{dy_{1}}{y_{1}^2}\Big). \end{align} $$

It remains to evaluate the three expressions in the large parentheses in (6.28). Firstly, from (3.37), we have

(6.29) $$ \begin{align} \sum_{n_{1}= 1}^{\infty} \ \frac{\zeta(w, \alpha, \psi_{(n_{1},0)})}{|n_{1}|^{2s-\alpha_{3}^{w} }} &= \zeta\left(1+ \alpha_{1}^{w}-\alpha_{2}^{w}\right)\zeta\left(1+ \alpha_{1}^{w}-\alpha_{3}^{w}\right) \sum_{n_{1}= 1}^{\infty} \ \frac{\sigma_{\alpha_{2}^{w}-\alpha_{3}^{w}}(|n_{1}|)}{|n_{1}|^{2s-\alpha_{3}^{w} }} \nonumber\\ &= \zeta\left(1+ \alpha_{1}^{w}-\alpha_{2}^{w}\right)\zeta\left(1+ \alpha_{1}^{w}-\alpha_{3}^{w}\right) \zeta\left(2s-\alpha_{3}^{w}\right)\zeta(2s-\alpha_{2}^{w}) \end{align} $$

for $\sigma> 1/2+\epsilon _{0}$ . Secondly, using (3.4) and the relation $\alpha _{1}^{w}+\alpha _{2}^{w}+\alpha _{3}^{w}=0$ , we have

(6.30) $$ \begin{align} \int_{0}^{\infty} \, W_{(\alpha_{2}^{w}-\alpha_{3}^{w})/2}(y_{0}) y_{0}^{2s-1+ (1+\alpha_{1}^{w})/2} \ \frac{dy_{0}}{y_{0}} = \frac{1}{2}\, \Gamma_{\mathbb{R}}(2s-\alpha_{3}^{w}) \Gamma_{\mathbb{R}}(2s- \alpha_{2}^{w}) \end{align} $$

for $\sigma> \epsilon _{0}$ . Thirdly, it follows from (3.19) and (3.12) that

(6.31) $$ \begin{align} 2\, \int_{0}^{\infty} \, h(y_{1}) \, \int_{0}^{1} \, E(z; s+ \alpha_{1}^{w} )e(-u) \ du\ \frac{dy_{1}}{y_{1}^2} = 2\, \frac{H(s+ \alpha_{1}^{w})}{\Lambda(1+2(s+\alpha_{1}^{w}))} \,. \end{align} $$

Now, from (6.28), (6.29), (6.30) and (6.31), it follows that $ \mathcal {J}_{(*,0)}^{\min }(s; \, h, \, \Phi )$ is given by

(6.32) $$ \begin{align} \sum_{w=w_{3}, w_{5}, w_{\ell}} \ H(s+ \alpha_{1}^{w}) \, \frac{ \Lambda(1+ \alpha_{1}^{w}-\alpha_{2}^{w})\Lambda(1+ \alpha_{1}^{w}-\alpha_{3}^{w}) \Lambda(2s+\alpha_{1}^{w}+ \alpha_{2}^{w})\Lambda(2s+\alpha_{1}^{w} +\alpha_{3}^{w})}{\Lambda(1+2(s+\alpha_{1}^{w}))}. \end{align} $$

Recalling the Weyl actions of $w_{3}, w_{5}, w_{\ell }$ on the parameters $(\alpha _{1}, \alpha _{2}, \alpha _{3})$ in (3.30) and replacing $\Phi \to \widetilde {\Phi }$ , the desired result follows.

The following result is immediate.

6.5 Diagonal and preparation for off-diagonal: $\mathcal {J}^{\mathrm {reg}}(s; \, h, \, \widetilde {\Phi })$

Proposition 6.13. On the vertical strip $1+\epsilon _{0}< \sigma < 4$ , we have

(6.33) $$ \begin{align} \mathcal{J}(s; \, h, \, \widetilde{\Phi}) &= \frac{ 1}{2}\, \prod_{i=1}^{3} \zeta(2s+\alpha_{i}) \ \int_{(0)} \, H(\mu)\, \prod_{i=1}^{3} \, \prod_{\pm} \Gamma_{\mathbb{R}}(s\pm \mu-\alpha_{i})\ d^{W}\mu + \mathrm{OD}_{\alpha}(s)\nonumber\\ &\quad + \mathcal{J}_{(0,1)}^{\min}(s; \, h, \, \widetilde{\Phi}) - \mathcal{J}_{(*,0)}^{\min}(s; \, h, \, \widetilde{\Phi}), \end{align} $$

where

(6.34) $$ \begin{align} \mathrm{OD}_{\alpha}(s) &= \lim_{\phi\to \pi/2} \frac{1}{4} \int_{(1+2\epsilon_{0})} \zeta\left(2s-s_{0}\right) \prod_{i=1}^{3} \, \zeta(s_{0}+\alpha_{i}) \, \sum_{\pm} (\mathcal{F}_{\alpha}^{\pm}H)(s_{0}, \, s; \, \phi) \ \frac{ds_{0}}{2\pi i}. \end{align} $$

Proof. Recall Proposition 6.6 on the decomposition of periods. From [Reference KwanKwa24, eq. (6.7)], we have

(6.35) $$ \begin{align} \mathcal{J}^{\mathrm{reg}}(s; \, h, \, \widetilde{\Phi}) = 2\cdot L(2s, \Phi) \int_{0}^{\infty} \int_{0}^{\infty} h(y_{1})(y_{0}^2 y_{1})^{s-\frac{1}{2}} W_{-\alpha}(y_{0}, y_{1}) \, d^{*}\mathbf{y} + \mathrm{OD}_{\alpha}(s), \end{align} $$

where the term $\mathrm {OD}_{\alpha }(s)$ denotes the ‘off-diagonal’, i.e., terms with $a_{0}a_{1}\neq 0$ in (6.13). The first term of (6.35) can be evaluated by (3.39), (3.13) and (3.10). By [Reference KwanKwa24, eq. (6.8)], we have

(6.36) $$ \begin{align} \mathrm{OD}_{\alpha}(s) &:= \sum_{a_{0}\neq 0} \sum_{a_{1}\neq 0} \ \frac{\mathcal{B}_{\Phi}(1,a_{1})}{|a_{0}|^{2s-1}|a_{1}|} \cdot \int_{0}^{\infty} \int_{0}^{\infty} h(y_{1})\, (y_{0}^2 y_{1})^{s-\frac{1}{2}} \, e\Big( \frac{a_{1}}{a_{0}}\, \frac{y_{0}^2}{1+y_{0}^2}\Big) \nonumber\\ & \quad \cdot W_{-\alpha}\Big( \big|\frac{a_{1}}{a_{0}}\big| \, \frac{y_{0}}{1+y_{0}^2}, \ y_{1} \sqrt{1+y_{0}^2} \Big) \ d^{*}\mathbf{y}, \end{align} $$

where the expression (6.36) converges absolutely whenever $1+\epsilon _{0}< \sigma < 4$ and $H \in \mathcal {C}_{\eta }$ ( $\eta>40$ ). By [Reference KwanKwa24, Proposition 7.2] and (3.39), this can be written as (6.34). This completes the proof.

6.6 Symmetries of the integral transform

The symmetries of $(\mathcal {F}_{\alpha }H)(s_{0}, s)$ play a crucial role in establishing the CFKRS conjecture for the cubic moment and are important for the analytic continuation argument in Section 6.7.

Proposition 6.14. The function $s\mapsto \left (\mathcal {F}_{\alpha }H\right )\left (1-\alpha _{1},s\right ) $ admits a holomorphic continuation to $\epsilon _{0}< \sigma < 4$ except for a simple pole at $s=(1-\alpha _{1})/2$ . Furthermore, we have

(6.37) $$ \begin{align} & (\mathcal{F}_{\alpha}H)(1-\alpha_{1}, \, s) \nonumber \\ & \quad = 2 \frac{\Gamma_{\mathbb{R}}( 2s-1+\alpha_{1})}{\Gamma_{\mathbb{R}}( 2-2s-\alpha_{1})} \int_{(0)} \, H(\mu) \, \prod_{\pm} \, \Gamma_{\mathbb{R}}( s\pm \mu-\alpha_{1})\prod_{j=2}^{3}\Gamma_{\mathbb{R}}(1-s\pm \mu+\alpha_{j}) \ d^{W}\mu \end{align} $$

on the strip $\epsilon _{0}< \sigma <1-\epsilon _{0}$ . The results for $\left (\mathcal {F}_{\alpha }H\right )\left (1-\alpha _{i},s\right )$ ( $i=2,3$ ) follow by symmetry.

Proof. The holomorphy of $s\mapsto \left (\mathcal {F}_{\alpha }H\right )\left (1-\alpha _{1},s\right ) $ on $\frac {1}{2}+\epsilon _{0} < \sigma < 4$ follows immediately from Proposition 4.1. Next, we consider a smaller domain $\frac {1}{2}+\epsilon _{0} < \sigma < 1-\epsilon _{0}$ . Put $s_{0}=1-\alpha _{1}$ . In (4.5), the factor $\Gamma \left (\frac {1-u}{2}\right )$ in the denominator cancels with $\Gamma \left ( \frac {s_{0}+\alpha _{1}-u}{2}\right )$ in the numerator, and hence,

(6.38) $$ \begin{align} & \left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right) \nonumber \\ &\quad = \pi^{\alpha_{1}-1/2} \ \int_{(\eta-1/2)} \ \widetilde{h}\big(s-s_{1}-\frac{1}{2}\big) \frac{\pi^{-s_{1}} \prod_{i=1}^{3} \ \Gamma\left( \frac{s_{1}-\alpha_{i}}{2}\right) }{ \Gamma\big(\frac{s_{1}-1}{2}+ \alpha_{1}\big)}\nonumber\\ & \quad\cdot \int_{(\epsilon_{0})} \frac{\Gamma(\frac{u}{2}) \Gamma\big(\frac{s_{1}+\alpha_{1}+u}{2}-s\big) \Gamma\big(\frac{1-\alpha_{1}+\alpha_{2}-u}{2}\big) \Gamma\big(\frac{1-\alpha_{1}+\alpha_{3}-u}{2}\big) \Gamma\big(s+\frac{-1+\alpha_{1}-u}{2}\big)}{\Gamma\big( \frac{1-\alpha_{1}+s_{1}-u}{2}\big)} \frac{du}{2\pi i} \frac{ds_{1}}{2\pi i}. \end{align} $$

We take $(a,b,c; d,e) := \big ( \frac {1-\alpha _{1}+\alpha _{2}}{2}, \ \frac {1-\alpha _{1}+\alpha _{3}}{2}, \ s-\frac {1-\alpha _{1}}{2}; \ 0, \ \frac {s_{1}+\alpha _{1}}{2}-s \big )$ , and verify that

$$ \begin{align*} (a+b)+c+d+e = \Big(1-\alpha_{1}- \frac{\alpha_{1}}{2}\Big) + s-\frac{1-\alpha_{1}}{2}+ \frac{s_{1}+\alpha_{1}}{2}-s = \frac{s_{1}-\alpha_{1}+1}{2} \ \ (:=f). \end{align*} $$

By Lemma 3.1 and a change of variables $u\to -2u$ , the u-integral in (6.38) is equal to

(6.39) $$ \begin{align} 2 \cdot \frac{\Gamma\big( \frac{1-\alpha_{1}+\alpha_{2}}{2}\big) \Gamma\big(\frac{1-\alpha_{1}+\alpha_{3}}{2}\big) \Gamma\big( s-\frac{1}{2}+ \frac{\alpha_{1}}{2}\big) \Gamma\big(\frac{s_{1}+1+\alpha_{2}}{2}-s\big) \Gamma\big( \frac{s_{1}+1+\alpha_{3}}{2}-s\big)\Gamma\big( \frac{s_{1}-1}{2}+\alpha_{1}\big)}{\Gamma\big( \frac{s_{1}-\alpha_{2}}{2}\big) \Gamma\big( \frac{s_{1}-\alpha_{3}}{2}\big) \Gamma\big( 1-\alpha_{1}-s +\frac{s_{1}}{2}\big)}. \end{align} $$

Now, observe that the factors $\Gamma ( \frac {s_{1}-\alpha _{2}}{2}) \Gamma ( \frac {s_{1}-\alpha _{3}}{2}) $ occur in both the denominator of (6.39) and in the numerator of the intergrand in (6.38). After making this cancellation, we have

(6.40) $$ \begin{align} \left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right) & = \Gamma\Big( \frac{1-\alpha_{1}+\alpha_{2}}{2}\Big) \Gamma\Big(\frac{1-\alpha_{1}+\alpha_{3}}{2}\Big) \Gamma\Big( s-\frac{1}{2}+ \frac{\alpha_{1}}{2}\Big) \nonumber\\ & \quad \cdot\, 2 \pi^{\alpha_{1}-1/2} \int_{(\eta-1/2)} \ \widetilde{h}\big(s-s_{1}-\frac{1}{2}\big) \nonumber\\ & \quad \cdot \frac{\pi^{-s_{1}} \Gamma\big( \frac{s_{1}-\alpha_{1}}{2}\big)\Gamma\big(\frac{s_{1}+1+\alpha_{2}}{2}-s\big) \Gamma\big( \frac{s_{1}+1+\alpha_{3}}{2}-s\big) }{\Gamma\big( 1-\alpha_{1}-s +\frac{s_{1}}{2}\big)} \ \frac{ds_{1}}{2\pi i}. \end{align} $$

Let $\sigma _{1}\in (2\sigma -1+\epsilon _{0},\, \sigma )$ . Using Lemma 3.11, we have

(6.41) $$ \begin{align} &\left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{1},s\right) \nonumber\\ & \quad = \pi^{\alpha_{1}-1/2-s} \ \Gamma\Big( \frac{1-\alpha_{1}+\alpha_{2}}{2}\Big) \Gamma\Big(\frac{1-\alpha_{1}+\alpha_{3}}{2}\Big) \Gamma\Big( s-\frac{1}{2}+ \frac{\alpha_{1}}{2}\Big) \int_{(0)} \ H(\mu) \nonumber\\ & \quad\cdot \int_{(\sigma_{1})} \frac{\Gamma\big( \frac{s_{1}-\alpha_{1}}{2}\big)\Gamma\big(\frac{s_{1}+1+\alpha_{2}}{2}-s\big) \Gamma\big( \frac{s_{1}+1+\alpha_{3}}{2}-s\big) \Gamma\big(\frac{s-s_{1}+\mu}{2}\big) \Gamma\big(\frac{s-s_{1}-\mu}{2}\big) }{\Gamma\big( 1-\alpha_{1}-s +\frac{s_{1}}{2}\big)} \frac{ds_{1}}{2\pi i} d^{W}\mu. \end{align} $$

For the $s_{1}$ -integral in (6.41), we apply the change of variables $s_{1}\to 2s_{1}$ , and Lemma 3.1 the second time but with $(a,b,c;d, e) := ( -\frac {\alpha _{1}}{2}, \ \frac {1+\alpha _{2}}{2}-s, \ \frac {1+\alpha _{3}}{2}-s; \ \frac {s+\mu }{2}, \ \frac {s-\mu }{2}).$ The $s_{1}$ -integral is now equal to

(6.42) $$ \begin{align} 2\cdot \frac{\Gamma\big( \frac{s+\mu-\alpha_{1}}{2}\big) \Gamma\big( \frac{s-\mu-\alpha_{1}}{2}\big) \Gamma\big( \frac{1+\alpha_{2}+\mu-s}{2}\big) \Gamma\big( \frac{1+\alpha_{2}-\mu-s}{2}\big) \Gamma\big( \frac{1+\alpha_{3}+\mu-s}{2}\big) \Gamma\big( \frac{1+\alpha_{3}-\mu-s}{2}\big)}{\Gamma\big(1-s-\frac{\alpha_{1}}{2}\big) \Gamma\big(\frac{1-\alpha_{2}}{2}-\alpha_{1}\big) \Gamma\big(\frac{1-\alpha_{3}}{2}-\alpha_{1}\big)}, \end{align} $$

upon observing that $a+(b+c)+d+e \, = \, -\frac {\alpha _{1}}{2} + ( 1-\frac {\alpha _{1}}{2}-2s) \, + s\, = \, 1-\alpha _{1}-s \, (:=f)$ .

The conclusion (6.37) follows from the cancellation of $\Gamma (\frac {1-\alpha _{2}}{2}-\alpha _{1}) \Gamma (\frac {1-\alpha _{3}}{2}-\alpha _{1})$ in the denominator of (6.42) and $ \Gamma ( \frac {1-\alpha _{1}+\alpha _{2}}{2}) \Gamma (\frac {1-\alpha _{1}+\alpha _{3}}{2}) $ in (6.41), since $\alpha _{1}+\alpha _{2}+\alpha _{3}=0$ . Finally, it is clear that (6.37) is holomorphic on $\epsilon _{0}<\sigma < 1-\epsilon _{0}$ except at $s=(1-\alpha _{1})/2$ . Combining this with the initial region of holomorphy, the continuation of $s\mapsto \left (\mathcal {F}_{\alpha }H\right )\left (1-\alpha _{1},s\right ) $ to $\epsilon _{0}< \sigma < 4$ is now established.

Proposition 6.15. The function $s\mapsto \left (\mathcal {F}_{\alpha }H\right )\left (2s-1, s \right ) $ admits a holomorphic continuation to $\epsilon _{0}<\sigma <4$ except for three simple poles at $s=(1-\alpha _{i})/2$ $(i=1,2,3)$ . On $\epsilon _{0} < \sigma < 1-\epsilon _{0}$ , we have

(6.43) $$ \begin{align} \left(\mathcal{F}_{\alpha}H\right)\left(2s-1, s \right) &= 2 \, \prod_{i=1}^{3} \frac{\Gamma_{\mathbb{R}}(2s-1+ \alpha_{i})}{\Gamma_{\mathbb{R}}(2-2s- \alpha_{i})} \, \int_{(0)} \, H(\mu)\, \prod_{i=1}^{3} \ \prod_{\pm} \, \Gamma_{\mathbb{R}}\big( 1-s+ \alpha_{i}\pm \mu\big) \ d^{W}\mu. \end{align} $$

Proof. The argument also appears in [Reference KwanKwa24]; for completeness, we include a proof here. Suppose $1/2+\epsilon _{0} < \sigma < 4$ and $s_{0}=2s-1$ . In (4.5), the factor $\Gamma \left (\frac {1-u}{2}\right )$ in the denominator cancels with $\Gamma \left (s-\frac {s_{0}+u}{2}\right )$ in the numerator. This leads to

(6.44) $$ \begin{align} \left(\mathcal{F}_{\alpha} H\right)\left(2s-1, s \right) & = \pi^{3/2-2s} \ \int_{(\eta-1/2)} \ \widetilde{h}\Big(s-s_{1}-\frac{1}{2}\Big) \frac{\pi^{-s_{1}} \prod_{i=1}^{3} \ \Gamma\big( \frac{s_{1}-\alpha_{i}}{2}\big) }{ \Gamma\big(\frac{1+s_{1}}{2}+1-2s\big)} \nonumber\\ & \quad \cdot \int_{(\epsilon_{0})} \ \frac{ \Gamma\left( \frac{u}{2}\right) \Gamma\left( \frac{u+s_{1}}{2}+ 1-2s\right) \cdot \prod_{i=1}^{3} \ \Gamma\left(s-\frac{1}{2}+ \frac{\alpha_{i}-u}{2}\right) }{ \Gamma\left(s-\frac{1}{2}+\frac{s_{1}-u}{2}\right)} \ \frac{du}{2\pi i} \ \frac{ds_{1}}{2\pi i}. \end{align} $$

Make the change of variables $u\to -2u$ and take $ (a,b,c )= (s-\frac {1}{2}+ \frac {\alpha _{1}}{2}, s-\frac {1}{2}+ \frac {\alpha _{2}}{2}, s-\frac {1}{2}+ \frac {\alpha _{3}}{2} )$ and $(d,e)= ( 0, \ \frac {s_{1}}{2}+1-2s)$ . By Lemma 3.1, the u-integral is equal to

(6.45) $$ \begin{align} 2 \, \prod_{i=1}^{3} \ \Gamma\Big(s-\frac{1}{2}+ \frac{\alpha_{i}}{2}\Big) \Gamma\Big(\frac{s_{1}+1+\alpha_{i}}{2}-s\Big)\Gamma\Big( \frac{s_{1}-\alpha_{i}}{2}\Big)^{-1}. \end{align} $$

Observe that the three $\Gamma $ -factors in the numerator of the integrand in (6.44) cancel with those in (6.45). Hence, we have

$$ \begin{align*} \left(\mathcal{F}_{\alpha} H\right)\left(2s-1, s \right) & = 2\pi^{3/2-2s} \, \prod_{i=1}^{3} \, \Gamma\big(s-\frac{1}{2}+ \frac{\alpha_{i}}{2}\big) \\ &\quad \times \int_{(\eta-1/2)} \ \widetilde{h}\big(s-s_{1}-\frac{1}{2}\big) \, \frac{\pi^{-s_{1}} \prod_{i=1}^{3} \Gamma\big(\frac{s_{1}+1+\alpha_{i}}{2}-s\big) }{ \Gamma\big(\frac{1+s_{1}}{2}+1-2s\big)} \ \frac{ds_{1}}{2\pi i}. \nonumber \end{align*} $$

We further restrict to $1/2+\epsilon _{0} < \sigma < 1-\epsilon _{0}$ . We shift the line of integration to the left from $\operatorname {\mathrm {Re}} s_{1}=\eta -1/2$ to $\operatorname {\mathrm {Re}} s_{1}=\sigma _{1} \in ( 2\sigma -1 +\epsilon _{0},\, \sigma )$ . No pole is crossed and we apply Lemma 3.11 to obtain

$$ \begin{align*} \left(\mathcal{F}_{\alpha} H\right)\left(2s-1, s \right) &= \pi^{\frac{3}{2}-3s} \, \prod_{i=1}^{3} \ \Gamma\big(s-\frac{1}{2}+ \frac{\alpha_{i}}{2}\big) \\ &\quad \times \int_{(0)} \, H(\mu) \int_{(\sigma_{1})} \, \frac{\prod_{i=1}^{3} \Gamma\big(\frac{s_{1}+1+\alpha_{i}}{2}-s\big) \,\prod_{\pm} \, \Gamma\big(\frac{s-s_{1}\pm\mu}{2}\big) }{\Gamma( \frac{1+s_{1}}{2}+1-2s)} \ \frac{ds_{1}}{2\pi i} \ d^{W}\mu. \nonumber \end{align*} $$

We apply the change of variable $s_{1}\to 2s_{1}$ , and Lemma 3.1 the second time, but with $(a,b, c) = (\frac {1}{2}-s+ \frac {\alpha _{1}}{2}, \ \frac {1}{2}-s+ \frac {\alpha _{2}}{2}, \ \frac {1}{2}-s+ \frac {\alpha _{3}}{2})$ and $ \left ( d,e\right ) = \left ( \frac {s+\mu }{2}, \ \frac {s-\mu }{2} \right )$ . The result follows as the $s_{1}$ -integral becomes

$$ \begin{align*} 2\ \prod_{i=1}^{3}\, \prod_{\pm} \ \Gamma\Big( \frac{1-s+ \alpha_{i}\pm \mu}{2}\Big)\Gamma\Big(1-s- \frac{\alpha_{i}}{2}\Big)^{-1}. \end{align*} $$

6.7 Analytic continuation and polar terms

The analytic continuation argument performed in [Reference KwanKwa24] is robust and carries over to the present case with minor modifications.

Proposition 6.16. The function $\mathrm {OD}_{\alpha }(s)$ admits a meromorphic continuation to the domain $1/4 + \epsilon _{0}\, < \, \sigma \, < \, 4$ . On the smaller domain $1/4 +\epsilon _{0}\, < \, \sigma \, < \, 3/4$ , the following equality holds:

(6.46) $$ \begin{align} 4\cdot \mathrm{OD}_{\alpha}(s) = & \sum_{i=1}^{3} \ \zeta(2s- 1+\alpha_{i}) \prod_{\substack{ 1\le j\le 3\\ j\neq i}}\zeta(1-\alpha_{i}+\alpha_{j}) \left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{i}, s\right) \end{align} $$

(6.47) $$ \begin{align} &\hspace{-5pt} + \prod_{i=1}^{3} \zeta(2s-1+\alpha_{i}) \left(\mathcal{F}_{\alpha}H\right)\left(2s-1, s\right) \end{align} $$

(6.48) $$ \begin{align} & \hspace{80pt} + \int_{(1/2)} \zeta(2s-s_{0}) \prod_{i=1}^{3}\, \zeta(s_{0}+\alpha_{i}) (\mathcal{F}_{\alpha}H)(s_{0}, s) \ \frac{ds_{0}}{2\pi i}. \end{align} $$

Proof. Readers are invited to consult [Reference KwanKwa24, Section 9] for fuller details. Parallel to [Reference KwanKwa24, Section 9A] (‘Step 1’), we shift the line of integration in (6.34) to $\operatorname {\mathrm {Re}} s_{0}=2\epsilon _{0}$ . This time, however, we pick up the residues of three simple poles at $s_{0} = 1-\alpha _{i} $ ( $i=1,2,3$ ), which results in

(6.49) $$ \begin{align} 4\cdot \mathrm{OD}_{\alpha}(s) = \lim_{\phi \to \pi/2} \ \bigg\{ & \sum_{i=1}^{3} \ \zeta(2s- 1+\alpha_{i}) \prod_{\substack{ 1\le j\le 3\\ j\neq i}}\zeta(1-\alpha_{i}+\alpha_{j}) \sum_{\pm} (\mathcal{F}_{\alpha}^{\pm}H)(1-\alpha_{i}, \, s; \, \phi) \nonumber\\ & \hspace{30pt} + \int_{(2\epsilon_{0})} \zeta\left(2s-s_{0}\right) \prod_{i=1}^{3}\, \zeta(s_{0}+\alpha_{i}) \sum_{\pm} (\mathcal{F}_{\alpha}^{\pm}H)(s_{0}, \, s; \, \phi) \ \ \frac{ds_{0}}{2\pi i} \,\bigg\}. \end{align} $$

This serves as a holomorphic continuation to $1/2+\epsilon _{0} < \sigma <4$ , except for the three simple poles at $s=1- \alpha _{i}/2$ ( $i=1,2,3$ ). Parallel to [Reference KwanKwa24, Section 9B] (‘Step 2’), we restrict to the strip $1/2+ \epsilon _{0} < \sigma < 3/4$ , and shift the line of integration from $\operatorname {\mathrm {Re}} s_{0}=2\epsilon _{0}$ to $\operatorname {\mathrm {Re}} s_{0}=1/2$ , crossing the simple pole of $\zeta (2s-s_{0})$ which has residue $-1$ . The function on the second line of (6.49) is holomorphic on the strip $1/4+\epsilon _{0}<\sigma < 3/4$ .

Section 9C of [Reference KwanKwa24] (‘Step 3’) carries over to the present context without change as it merely takes care of the necessary regularity (using Proposition 4.1 and the imposed assumptions on our class of test functions). Section 9D of [Reference KwanKwa24] (‘Step 4’) concerns the continuation of (6.46) and (6.47). Here, we instead adopt an explicit approach and the desired conclusion follows from Proposition 6.14–6.15. This completes the proof.Footnote ¹²

6.8 Agreement with CFKRS and completion the Proof of Theorem 1.3

By Lemma 6.2, (6.7), and (6.9), we have

(6.50) $$ \begin{align} \mathfrak{M}_{-\alpha}^{(3)}(s; H) = 2 \cdot \mathcal{J}(s; \, h, \, \widetilde{\Phi}) \end{align} $$

on the vertical strip $3/2< \sigma < 4$ . By Corollaries 6.4 and 6.17, both sides of (6.50) admit a meromorphic continuation to the strip $1/4+\epsilon _{0}<\sigma < 4$ , and (6.50) remains valid on the new strip. Now, we restrict to $1/4 + \epsilon _{0} < \sigma < 3/4$ . Putting Corollary 6.4, (6.33), (6.24), (6.25), (6.46), (6.47), (6.48) together, we have

(6.51) $$ \begin{align} & \mathfrak{M}_{-\alpha}^{(3)}(s; H) + \mathcal{R}_{-\alpha}(s; H) + \mathcal{R}_{\alpha}(1-s; H)\hspace{140pt} \end{align} $$

(6.52) $$ \begin{align} & = \prod_{i=1}^{3}\, \zeta(2s+\alpha_{i}) \int_{(0)} \, H(\mu) \, \prod_{i=1}^{3} \, \prod_{\pm} \, \Gamma_{\mathbb{R}}(s\pm \mu-\alpha_{i}) \ d^{W}\mu \ \ + \mathcal{M}_{-\alpha}^{1}(s; H) \end{align} $$

(6.53) $$ \begin{align} & + \frac{1}{2} \ \sum_{i=1}^{3} \ \zeta(2s- 1+\alpha_{i}) \prod_{\substack{ 1\le j\le 3\\ j\neq i}}\zeta(1-\alpha_{i}+\alpha_{j}) \left(\mathcal{F}_{\alpha}H\right)\left(1-\alpha_{i}, s\right) \end{align} $$

(6.54) $$ \begin{align} & + \frac{1}{2} \, \prod_{i=1}^{3}\, \zeta(2s-1+\alpha_{i}) \left(\mathcal{F}_{\alpha} H\right)\left(2s-1, s\right)\hspace{90pt} \end{align} $$

(6.55) $$ \begin{align} & + \frac{1}{2} \ \int_{(1/2)} \zeta\left(2s-s_{0}\right)\ \prod_{i=1}^{3} \ \zeta(s_{0}+\alpha_{i}) \left(\mathcal{F}_{\alpha}H\right)\left(s_{0}, s\right) \ \frac{ds_{0}}{2\pi i}. \hspace{24pt} \end{align} $$

We complete the proof of Theorem 1.3 by showing that the terms (6.52), (6.53) and (6.54) agree with the CFKRS prediction for the $\mathrm {SO}(\text {even})$ symmetry described in Section 5, namely, their sum is precisely $\sum _{0\le i\le 3} \, \mathcal {M}_{-\alpha }^{i}(s; H)$ as defined in Theorem 1.3. Indeed, observe that:

1. (1) (0 & 1-swap). By $\alpha _{1}+\alpha _{2}+\alpha _{3}=0$ , the first term of (6.52) can be written as $\mathcal {M}_{-\alpha }^{0}(s; H)$ as defined by (1.9). The second term of (6.52) is already in the form of the $1$ -swap prediction.

2. (2) (2-swap). By Proposition 6.14 and the functional equation of the $\zeta $ -function of the form

   (6.56) $$ \begin{align} \zeta(2s-1+ \alpha_{i}) \frac{\Gamma_{\mathbb{R}}( 2s-1+\alpha_{i})}{\Gamma_{\mathbb{R}}( 2-2s-\alpha_{i})} = \zeta(2-2s-\alpha_{i}) = \zeta(2-2s + \sum_{\substack{1\le j\le 3\\ j\neq i}}\alpha_{j}), \end{align} $$
   it follows that (6.53) coincides with $\mathcal {M}_{-\alpha }^{2}(s; H)$ as defined by (1.11).

3. (3) (3-swap). Using Proposition 6.15 and (6.56) thrice, observe that (6.54) coincides with $\mathcal {M}_{-\alpha }^{3}(s; H)$ as defined by (1.12).

We set $s=1/2$ . The matching with the predictions in Section 5.1 follows from the actions of the Weyl group (3.30). The restriction $\alpha _{1}+\alpha _{2}+\alpha _{3}=0$ can be removed by analytic continuation in the parameters $\alpha _{i}$ ’s.

7.1 Agreement with the unitary CFKRS

To align with the set-up of Section 5.2, we re-label the dual moment of Theorem 1.3 as

(7.1) $$ \begin{align} \int_{(\frac{1}{2}+\gamma)} \ \zeta(2s-s_{0})\zeta(s_{0}+\nu_{1}) \zeta(s_{0}+\nu_{2}) \zeta(s_{0}+\nu_{3}) \left(\mathcal{F}_{\nu}H\right)\left(s_{0}, s\right) \ \frac{ds_{0}}{2\pi i}. \end{align} $$

Notice that the line of integration is shifted by a small quantity $\gamma $ . We put

(7.2) $$ \begin{align} s \, := \, \frac{1+\delta}{2}, \end{align} $$

and apply the functional equation (2.3) for the $\zeta $ -function to the factor $ \zeta (s_{0}+\nu _{3})$ , the moment (7.1) takes the form:

$$ \begin{align*} \frac{1}{2\pi} \ \int_{\mathbb{R}} \ \zeta\Big(\frac{1}{2}+\delta-\gamma-it\Big) &\zeta\Big(\frac{1}{2}-\gamma-\nu_{3}-it\Big) \prod_{j=1}^{2} \, \zeta\Big(\frac{1}{2}+\gamma+\nu_{j}+it\Big) \nonumber\\ &\hspace{20pt} \ \cdot \ \frac{\Gamma_{\mathbb{R}}(1/2-\gamma-\nu_{3}-it)}{\Gamma_{\mathbb{R}}(1/2+\gamma+\nu_{3}+it)} \ \left(\mathcal{F}_{\nu}H\right)\Big(\frac{1}{2}+\gamma+it,\ \frac{1+\delta}{2}\Big) \, dt.\nonumber \end{align*} $$

Following the setting of Section 5.2, we introduce the linear re-parametrizations:

$$ \begin{align*} \begin{cases} \alpha_{1} = \gamma + \nu_{1}\\ \alpha_{2} = \gamma + \nu_{2}\\ \beta_{1} = -\gamma - \nu_{3} \\ \beta_{2} = \delta - \gamma, \end{cases} \end{align*} $$

where $\sum \nu _{j}=0$ . We solve for $\gamma $ using the first three equations: $\gamma = (\alpha _{1} + \alpha _{2} - \beta _{1} )/3$ . It follows that

(7.3) $$ \begin{align} \begin{cases} \nu_{1} = (2\alpha_{1} - \alpha_{2} + \beta_{1})/3\\ \nu_{2} = (2 \alpha_{2} -\alpha_{1} + \beta_{1})/3\\ \nu_{3} = -(\alpha_{1}+\alpha_{2}+2\beta_{1})/3 \\ 3\delta = 3\beta_{2} + \alpha_{1} + \alpha_{2} - \beta_{1}. \end{cases} \end{align} $$

As a result, the moment (7.1) can be expressed as

(7.4) $$ \begin{align} \int_{\mathbb{R}} \, \zeta\Big( \frac{1}{2}+ \alpha_{1}+it\Big) &\zeta\Big( \frac{1}{2}+ \alpha_{2}+it\Big) \zeta\Big( \frac{1}{2}+ \beta_{1}-it\Big) \zeta\Big( \frac{1}{2}+ \beta_{2}-it\Big) \eta_{\alpha_{1}, \alpha_{2}; \beta_{1}, \beta_{2}}(t) \, dt, \end{align} $$

where the weight function is given by

$$ \begin{align*} & \eta_{\alpha_{1}, \alpha_{2}; \beta_{1}, \beta_{2}}(t) \\ & \quad := \frac{1}{2\pi} \frac{\Gamma_{\mathbb{R}}(1/2+\beta_{1}-it)}{\Gamma_{\mathbb{R}}(1/2-\beta_{1}+it)} \left(\mathcal{F}_{\nu}H\right)\Big(\frac{1}{2}+\frac{\alpha_{1} + \alpha_{2} - \beta_{1} }{3}+it, \frac{1}{2}+ \frac{\alpha_{1}+ \alpha_{2}-\beta_{1}+3\beta_{2}}{6}\Big). \end{align*} $$

We write $ \mathcal {R}_{\pm \alpha }(s; H) = \mathcal {R}_{\pm \nu }(s; H)$ (see (1.13)). Using the re-labeling (7.3) and repeated applications of the functional equation for $\Lambda (s)$ , it follows that $(1/2)\,\mathcal {R}_{\nu }(1-s; H)+ (1/2) \, \mathcal {R}_{-\nu }(s; H)$ is given by

$$ \begin{align*} & H\Big(\frac{1+\alpha_{1} - \alpha_{2} + \beta_{1} -\beta_{2}}{2}\Big) \frac{\Lambda(1+\alpha_{1}-\alpha_{2})\Lambda(1+\beta_{1}-\beta_{2})\Lambda(1+\alpha_{1}+\beta_{1}) \Lambda(1-\alpha_{2}-\beta_{2})}{\Lambda(2+\alpha_{1}-\alpha_{2}+\beta_{1}-\beta_{2})} \nonumber\\ + &\ H\Big(\frac{1-\alpha_{1}+\alpha_{2}+\beta_{1}-\beta_{2}}{2}\Big) \frac{\Lambda(1-\alpha_{1}+\alpha_{2}) \Lambda\left(1+\beta_{1}-\beta_{2}\right)\Lambda(1+\alpha_{2}+\beta_{1}) \Lambda(1-\alpha_{1}-\beta_{2}) }{\Lambda(2-\alpha_{1}+\alpha_{2}+\beta_{1}-\beta_{2})} \nonumber\\ + &\ H\Big(\frac{1-\alpha_{1}-\alpha_{2}-\beta_{1}-\beta_{2}}{2}\Big) \frac{\Lambda(1-\alpha_{1}-\beta_{1})\Lambda(1-\alpha_{2}-\beta_{2}) \Lambda(1-\alpha_{2}-\beta_{1})\Lambda(1-\alpha_{1}-\beta_{2})}{\Lambda(2-\alpha_{1}-\alpha_{2}-\beta_{1}-\beta_{2})}, \nonumber\\ + &\ H\Big(\frac{1-\alpha_{1}+\alpha_{2}-\beta_{1}+\beta_{2}}{2}\Big) \frac{\Lambda(1-\alpha_{1}-\beta_{1}) \Lambda(1-\alpha_{1}+\alpha_{2})\Lambda(1+\alpha_{2}+\beta_{2})\Lambda(1-\beta_{1}+\beta_{2})}{\Lambda(2-\alpha_{1}+\alpha_{2}-\beta_{1}+\beta_{2})} \nonumber\\ + &\ H\Big(\frac{1+\alpha_{1}-\alpha_{2}-\beta_{1}+\beta_{2}}{2}\Big) \frac{\Lambda(1-\alpha_{2}-\beta_{1})\Lambda(1+\alpha_{1}-\alpha_{2})\Lambda(1+\alpha_{1}+\beta_{2}) \Lambda(1-\beta_{1}+\beta_{2})}{\Lambda(2+\alpha_{1}-\alpha_{2}-\beta_{1}+\beta_{2})} \nonumber\\ + &\ H\Big(\frac{1+\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2}}{2}\Big) \frac{\Lambda(1+\alpha_{2}+\beta_{1})\Lambda(1+\alpha_{1}+\beta_{1})\Lambda(1+\alpha_{1}+\beta_{2})\Lambda(1+\alpha_{2}+\beta_{2})}{\Lambda(2+\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2})}. \nonumber \end{align*} $$

We now readily observe that the quotients and products of $\zeta $ ’s above match exactly with those in the CFKRS conjecture (5.6)! Moreover, the cubic moment dual to (7.4) takes the form:

$$ \begin{align*} \sum_{j} \frac{H(\mu_{j})}{\langle \, \phi_{j}, \phi_{j}\, \rangle} & \Lambda\big( \frac{1-\alpha_{1} + \alpha_{2} - \beta_{1} +\beta_{2}}{2}, \, \phi_{j}\big) \\ &\Lambda\big( \frac{1-\alpha_{2} + \alpha_{1} - \beta_{1} +\beta_{2}}{2} , \,\phi_{j}\big) \Lambda\big(\frac{1+\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2}}{2}, \,\phi_{j}\big) \end{align*} $$

plus the corresponding continuous contribution.

7.2 Comments

In the literature, moments of L-functions are more commonly approached with approximate functional equations. A clear advantage is that one can take $s=1/2$ right from the start. For example, tracing the arguments of [Reference PetrowPet15] for the cubic moment of $\text {GL}(2)\ L$ -functions (readily adaptable to the Maass case and closely related to [Reference YoungYou17]), the main terms for the fourth moment of the $\text {GL}(1)\ L$ -functions do not appear: most likely, they are small and absorbed into the error term of the approximate Motohashi-type formula obtained in [Reference PetrowPet15]. In other words, the ‘approximate’ treatment precludes the possibility of spectral inversion by changing the test vectors (see the end of Section 1.5). This is not satisfactory, as one should be able to pass between the two different-looking moments in a Motohashi-type formula for distinct applications, much as with the celebrated Kuznetsov formulae.

In Section 1.4, we compare the sources of the main terms in the fourth moment with earlier methods. For the cubic moment, previous works of [Reference IvićIvi02] and [Reference FrolenkovFro20] rely on the shifted divisor sums to extract the main terms. The work [Reference FrolenkovFro20] (in the weight aspect) is closer in spirit to [Reference Conrey, Farmer, Keating, Rubinstein and SnaithCFK+05], which begins with an exact identity for the twisted second moment of $\text {GL}(2)\ L$ -functions, obtained quite non-trivially from the Petersson formula. The dual side of the moment identity comprises three pieces of shifted divisor sums. In total, $12$ different residues arise from these sums. As in [Reference Hughes and YoungHY10, Reference YoungYou11], delicate combinations of terms are required, and [Reference FrolenkovFro20] carries this out. Interestingly, some combinations contribute to main terms, some to error terms, and some to a mixture of both! This appears to be the primary reason [Reference FrolenkovFro20] arrives at an approximate cubic moment identity, despite starting with an exact moment identity. The approaches of [Reference MotohashiMot93, Reference Bruggeman and MotohashiBM05, Reference NelsonNel20, Reference WuWu22, Reference Balkanova, Frolenkov and WuBFW24] and the present work do not encounter this issue. This phenomenon deserves further investigation.