A Theta–Regularized Identity for SL₂ and a Fejér–Windowed Strip Bridge for Log|ξ|

1. Introduction and Main Results

Let $\Gamma ={\mathrm{PSL}}_2\left(\mathbb{Z}\right)$ and $X=\Gamma \setminus \mathbb{H}$ be the modular surface, with $z=x+\mathrm{i}y$, $y>0$, and hyperbolic measure $d\mu \left(z\right)=y^{-2}dxdy$. Let $E(z,s)$ denote the spherical Eisenstein series at the cusp ∞ for $G={\mathrm{SL}}_2\left(\mathbb{R}\right)$, normalized so that its constant term at ∞ is

$$E(z,s)=y^s+\varphi \left(s\right)y^{1-s}+\dots ,$$

(1)

where $\varphi \left(s\right)$ is the scattering coefficient. For the standard normalization (see, e.g., [20]),

$$\varphi \left(s\right)={\displaystyle \frac{\xi (2s-1)}{\xi \left(2s\right)}}={\pi }^{{\displaystyle \frac{1}{2}}}{\displaystyle \frac{\Gamma \left(s-\frac{1}{2}\right)}{\Gamma \left(s\right)}}{\displaystyle \frac{\zeta (2s-1)}{\zeta \left(2s\right)}},$$

(2)

with $\xi \left(s\right)={\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$ the completed Riemann zeta function.

The real–analytic Jacobi theta kernel is

$$\theta \left(z\right)=\sum _{m,n\in \mathbb{Z}}exp\left(-\pi {\displaystyle \frac{{|mz+n|}^2}{y}}\right),z=x+\mathrm{i}y,$$

(3)

and we write $\Theta \left(z\right):=\theta \left(z\right)-1$. Since $\Theta$ is x–periodic and grows like $\sqrt{y}$ in the cusp (see §Section 2.2), one must employ the regularized inner products in the sense of Zagier and Arthur for pairings involving $\Theta$ and Eisenstein series.

1.1. Theta–Weighted Mixed Energy and the Main Identity

Let $s=\frac{1}{2}+\sigma +\mathrm{i}t$ with $\sigma ,t\in \mathbb{R}$. With the standard $L^2$ pairing $\langle f,g\rangle ={\int }_{X}f\left(z\right)\overline{g\left(z\right)}d\mu \left(z\right)$ and its Zagier–Arthur regularized extension, consider the mixed theta–weighted energy

$${\mathcal{J}}_{\Theta }\left(s\right):={〈\Theta (·)E(·,s),\Theta (·)E(·,1-\overline{s})〉}_{\mathrm{reg}}.$$

(4)

The choice $s^{\prime }=1-\overline{s}$ ensures compatibility with $\xi (1-\overline{s})=\overline{\xi \left(s\right)}$ and produces a nontrivial t–dependence.

Our first main result is an explicit identity for the $\sigma$–derivative of $log|{\mathcal{J}}_{\Theta }\left(s\right)|$.

Theorem 1 

(Theta–regularized identity). Fix Tamagawa measures and spherical local data. For all $\sigma ,t\in \mathbb{R}$, in the sense of tempered distributions in t and pointwise away from zeros and poles, one has

$${\displaystyle \frac{\partial }{\partial \sigma }}log\left|{\mathcal{J}}_{\Theta }\left(\frac{1}{2}+\sigma +\mathrm{i}t\right)\right|=2mathrmRe{\displaystyle \frac{{\xi }^{\prime }(\frac{1}{2}+\sigma +\mathrm{i}t)}{\xi (\frac{1}{2}+\sigma +\mathrm{i}t)}}-2\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }(1+2\sigma +2\mathrm{i}t)}{\zeta (1+2\sigma +2\mathrm{i}t)}}+2\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }(1-2\sigma +2\mathrm{i}t)}{\zeta (1-2\sigma +2\mathrm{i}t)}}.$$

(5)

Equivalently,

$${\displaystyle \frac{\partial }{\partial \sigma }}log|{\mathcal{J}}_{\Theta }\left(s\right)|=2{\displaystyle \frac{\partial }{\partial \sigma }}log\left|\xi \left(s\right)\right|-{\displaystyle \frac{\partial }{\partial \sigma }}\mathrm{Re}log\zeta \left(2s\right)-{\displaystyle \frac{\partial }{\partial \sigma }}\mathrm{Re}log\zeta (2-2\overline{s}),$$

with $s={\displaystyle \frac{1}{2}}+\sigma +\mathrm{i}t$.

Remark 1. 

On the critical line $\sigma =0$ one has $2s=2-2\overline{s}=1+2\mathrm{i}t$, so the last two terms in (5) cancel identically and

$${\displaystyle \frac{\partial }{\partial \sigma }}log\left|{\mathcal{J}}_{\Theta }\left(\frac{1}{2}+\sigma +\mathrm{i}t\right)\right|{|}_{\sigma =0}=2{\displaystyle \frac{\partial }{\partial \sigma }}log\left|\xi \left(\frac{1}{2}+\sigma +\mathrm{i}t\right)\right|{|}_{\sigma =0}.$$

Thus the theta–regularized mixed energy has a particularly transparent logarithmic derivative along the critical line.

Remark 2 

(Growth of logarithmic derivatives and temperedness). Standard estimates (see, for example, [14] or [41]) show that for any fixed compact interval $\sigma \in [{\sigma }_1,{\sigma }_2]$,

$${\displaystyle \frac{{\xi }^{\prime }\left(\frac{1}{2}+\sigma +\mathrm{i}t\right)}{\xi \left(\frac{1}{2}+\sigma +\mathrm{i}t\right)}},{\displaystyle \frac{{\zeta }^{\prime }(1+2\sigma +2\mathrm{i}t)}{\zeta (1+2\sigma +2\mathrm{i}t)}},{\displaystyle \frac{{\zeta }^{\prime }(1-2\sigma +2\mathrm{i}t)}{\zeta (1-2\sigma +2\mathrm{i}t)}}=O\left(log\left(\right|t|+2)\right)$$

as $\left|t\right|\to \infty$. In particular, each term on the right–hand side of (5) defines a tempered distribution in t, justifying the distributional interpretation of 1.

1.2. Fejér–Windowed Version

The Fourier transform convention $\widehat{f}\left(\lambda \right)={\int }_{\mathbb{R}}f\left(u\right){\mathrm{e}}^{-\mathrm{i}\lambda u}du$ is used. Let $K_H$ denote the Fejér kernel

$$K_H\left(u\right):={\displaystyle \frac{H}{2\pi }}{\left({\displaystyle \frac{sin(Hu/2)}{Hu/2}}\right)}^2,{\widehat{K}}_H\left(\lambda \right)={\left(1-{\displaystyle \frac{\left|\lambda \right|}{H}}\right)}_{+}.$$

(6)

Then $K_H\in L^1\left(\mathbb{R}\right)$, ${\int }_{\mathbb{R}}{K}_H\left(u\right)du=1$, and ${\widehat{K}}_H$ is supported in $[-H,H]$.

Theorem 2 

(Fejér–windowed theta identity). Let $H\ge 1$ and $K_H$ be as in (6). For any ${\sigma }^{★},t_0\in \mathbb{R}$,

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial }{\partial \sigma }}{\int }_{\mathbb{R}}{K}_H(t-t_0)log\left|{\mathcal{J}}_{\Theta }\left(\frac{1}{2}+\sigma +\mathrm{i}t\right)\right|dt{|}_{\sigma ={\sigma }^{★}}& =2{\int }_{\mathbb{R}}{K}_H(t-t_0)\mathrm{Re}{\displaystyle \frac{{\xi }^{\prime }(\frac{1}{2}+{\sigma }^{★}+\mathrm{i}t)}{\xi (\frac{1}{2}+{\sigma }^{★}+\mathrm{i}t)}}dt\hfill \\ \hfill & -2{\int }_{\mathbb{R}}{K}_H(t-t_0)\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }(1+2{\sigma }^{★}+2\mathrm{i}t)}{\zeta (1+2{\sigma }^{★}+2\mathrm{i}t)}}dt\hfill \\ \hfill & +2{\int }_{\mathbb{R}}{K}_H(t-t_0)\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }(1-2{\sigma }^{★}+2\mathrm{i}t)}{\zeta (1-2{\sigma }^{★}+2\mathrm{i}t)}}dt.\hfill \end{array}$$

All integrals are to be interpreted as pairings with tempered distributions and are valid pointwise away from zeros and poles.

1.3. Fejér–Windowed Strip Bridge for $log\left|\xi \right|$

An independent harmonic analysis argument is used to prove a “strip bridge” for $u(\sigma ,t)=log\left|\xi \right(1/2+\sigma +\mathrm{i}t\left)\right|$, expressing the short–band component of the interior normal derivative ${\partial }_{\sigma }u$ in terms of boundary data.

Fix $h\in (0,1/2]$ and set

$$u(\sigma ,t):=log\left|\xi \left(\frac{1}{2}+\sigma +\mathrm{i}t\right)\right|,F_{\pm }\left(t\right):={\partial }_{\sigma }u(\pm h,t).$$

Write $F_{\mathrm{ev}}=\frac{1}{2}(F_{+}+F_{-})$ and $F_{\mathrm{odd}}=\frac{1}{2}(F_{+}-F_{-})$. Fix a short Fourier band

$$B_{{\lambda }_0,\Delta }=\{\lambda \in \mathbb{R}:|\lambda \pm {\lambda }_0|\le 2\Delta \},{\lambda }_0\asymp T,\Delta =T^{-1-\epsilon },$$

with small fixed $\epsilon >0$, and let $P_{\Delta }$ be the corresponding frequency projection.

Theorem 3 

(Fejér–windowed strip bridge). Fix ${\sigma }_0\in (0,h)$ and let $H\asymp T$ be chosen so that $|{\lambda }_0|\le (1-c)H$ for some fixed $c\in (0,1)$, hence $B_{{\lambda }_0,\Delta }\subset [-H,H]$. For any fixed ${\sigma }^{★}\in [-h,h]$ with $|{\sigma }^{★}|\ge {\sigma }_0$ and any $t_0\in \mathbb{R}$ with $|t_0|\le CH$, one has

$$\begin{array}{cc}\hfill {\int }_{\mathbb{R}}{K}_H(t-t_0)\left(P_{\Delta }{\partial }_{\sigma }u\right)({\sigma }^{★},t)dt& ={\tilde{m}}_{\mathrm{ev}}({\lambda }_0;{\sigma }^{★})(K_H*P_{\Delta }{F}_{\mathrm{ev}})\left(t_0\right)\hfill \\ \hfill & +{\tilde{m}}_{\mathrm{odd}}({\lambda }_0;{\sigma }^{★})(K_H*P_{\Delta }{F}_{\mathrm{odd}})\left(t_0\right)+O\left(H^{-\eta }\right),\hfill \end{array}$$

(7)

uniformly for $|t_0|\le CH$, where the frozen strip Poisson multipliers are

$${\tilde{m}}_{\mathrm{ev}}({\lambda }_0;{\sigma }^{★})={\displaystyle \frac{cosh\left({\lambda }_0{\sigma }^{★}\right)}{cosh\left({\lambda }_{0}h\right)}},{\tilde{m}}_{\mathrm{odd}}({\lambda }_0;{\sigma }^{★})={\displaystyle \frac{sinh\left({\lambda }_0{\sigma }^{★}\right)}{sinh\left({\lambda }_{0}h\right)}},$$

and $\eta >0$ depends only on ε. The error $O\left(H^{-\eta }\right)$ arises from short–band freezing and from local renormalization of nearby poles and zeros, after Fejér smoothing and application of $P_{\Delta }$, uniformly for $|{\sigma }^{★}|\ge {\sigma }_0$.

Remark 3 

(Dependence on the horizontal zero geometry). In the proof of Theorem 3, the renormalization error arising from $R_{t_0}^{\left(1\right)}$ admits the explicit bound

$${\int }_{\mathbb{R}}{K}_H(t-t_0)\left(P_{\Delta }{R}_{t_0}^{\left(1\right)}\right)({\sigma }^{★},t)dt\ll \sum _{\rho :|\Im \rho -t_0|\le 2H}{m}_{\rho }{\mathrm{e}}^{-cH|{\sigma }^{★}-(\Re \rho -\frac{1}{2})|},$$

where the sum runs over the zeros and the pole of ξ, counted with multiplicity $m_{\rho }$, and $c>0$ depends only on h. Combining this with $N\left(T\right)\ll TlogT$ and the band width $\Delta =T^{-1-\epsilon }$ yields the uniform bound $O\left(H^{-\eta }\right)$, for some $\eta >0$ depending only on ε. Under the Riemann Hypothesis, $\Re \rho =\frac{1}{2}$ for all zeros, so $|{\sigma }^{★}-(\Re \rho -\frac{1}{2})|=|{\sigma }^{★}|\ge {\sigma }_0$, and the sum is $\ll {\mathrm{e}}^{-c^{\prime }{\sigma }_{0}H}$; the error is then exponentially small in H (hence $O\left(H^{-A}\right)$ for every $A>0$).

2. Background: Eisenstein Series, Maass–Selberg, Theta Kernel, Weil Representation

2.1. Eisenstein Series and Maass–Selberg

Let $G={\mathrm{SL}}_2$, let B denote the upper Borel, and let $I\left(s\right)={\mathrm{Ind}}_{B\left(\mathbb{A}\right)}^{G\left(\mathbb{A}\right)}{\left(\right|·|}^s)$. Let $f_s={\otimes }_v{f}_{s,v}$ be the spherical section with $f_{s,\infty }\left(g_z\right)=y^s$ on $g_z=n\left(x\right)a\left(y\right)$, and $f_{s,v}\left(k_v\right)=1$ at finite v. Then

$$E(g,s):=\sum _{\gamma \in B\left(\mathbb{Q}\right)\setminus G\left(\mathbb{Q}\right)}{f}_s\left(\gamma g\right)$$

defines an Eisenstein series convergent for $\Re s>1$ and meromorphic in s. On X, $E(z,s)=E(g_z,s)$ has the Fourier expansion

$$E(z,s)=y^s+\varphi \left(s\right)y^{1-s}+\sum _{n\ne 0}{a}_n\left(s\right)\sqrt{y}{K}_{s-{\displaystyle \frac{1}{2}}}\left(2\pi \right|n\left|y\right)e\left(nx\right),$$

with $\varphi \left(s\right)$ as in (2).

The Maass–Selberg relation (see e.g. [18,20]) gives, for the truncated domain ${\mathcal{F}}_Y$,

$${\int }_{{\mathcal{F}}_Y}{\left|E(z,\frac{1}{2}+\mathrm{i}t)\right|}^{2}d\mu \left(z\right)=c_{0}logY+c_1\left(t\right)+R(Y,t),$$

(8)

where $c_0>0$ is absolute, $c_1\left(t\right)$ is bounded in t and expressible in terms of ${\varphi }^{\prime }(1/2+\mathrm{i}t)/\varphi (1/2+\mathrm{i}t)$, and $R(Y,t)=O\left(Y^{-c}\right)$ uniformly in t for some $c>0$.

2.2. Theta Kernel and Cusp Asymptotics

For $\theta \left(z\right)$ as in (3), Poisson summation in n yields

$$\theta \left(z\right)=\sqrt{y}\sum _{m,k\in \mathbb{Z}}{e}^{-\pi (m^2+k^2)y}{e}^{2\pi \mathrm{i}kmx}.$$

As $y\to \infty$, only $(m,k)=(0,0)$ contributes significantly, hence

$$\theta \left(z\right)=\sqrt{y}+O\left(\sqrt{y}{e}^{-\pi y}\right),\Theta \left(z\right)=\sqrt{y}-1+O\left(\sqrt{y}{e}^{-\pi y}\right).$$

(9)

Thus $\Theta \left(z\right)$ grows like $\sqrt{y}$ in the cusp.

2.3. Weil Representation and the Theta Lift

Let V be the split binary quadratic space over $\mathbb{Q}$ with $q\left(x\right)=x_1{x}_2$. In the Schrödinger model of the Weil representation $\omega$ for $(G,\mathrm{O}(V\left)\right)$ on $\left(V\right(\mathbb{A}\left)\right)$, for $\phi ={\otimes }_v{\phi }_v$ define

$$\theta (g,h;\phi ):=\sum _{x\in V\left(\mathbb{Q}\right)}\omega (g,h)\phi \left(x\right).$$

With ${\phi }_p={\mathbf{1}}_{V\left({\mathbb{Z}}_p\right)}$ for $p<\infty$ and ${\phi }_{\infty }\left(x\right)=e^{-\pi (x_1^2+x_2^2)}$, one verifies that $\theta (g_z,1;\phi )=\theta \left(z\right)$ and hence the regularized theta lift (in the sense of Kudla–Rallis and Ichino)

$${\Theta }_{\phi }\left(g\right):={\int }_{\mathrm{O}\left(V\right)\left(\mathbb{Q}\right)\setminus \mathrm{O}\left(V\right)\left(\mathbb{A}\right)}^{\mathrm{reg}}\theta (g,h;\phi )dh$$

satisfies ${\Theta }_{\phi }\left(g_z\right)=\Theta \left(z\right)$.

3. Local Doubling Zeta Integrals and Global Completion

Let $f_s$ be as in §Section 2.1. The local doubling zeta integrals are

$$Z_v(s,{\phi }_v,f_{s,v}):={\int }_{G\left({\mathbb{Q}}_v\right)}{f}_{s,v}\left(g\right)\langle {\omega }_v\left(g\right){\phi }_v,{\phi }_v\rangle dg.$$

For $p<\infty$ unramified, $Z_p(s,{\phi }_p,f_{s,p})={\zeta }_p\left(s\right)$, and for $v=\infty$ there is a (unique up to scalar) spherical choice of $f_{s,\infty }$ for which $Z_{\infty }(s,{\phi }_{\infty },f_{s,\infty })={\pi }^{-s/2}\Gamma (s/2)$ (see [56,58,60] and Appendix B). Thus, with Tamagawa measures and compatible local data,

$$\prod _v{Z}_v(s,{\phi }_v,f_{s,v})=\xi \left(s\right).$$

4. Rallis Inner Product Formula and the Theta Identity

For $s,s^{\prime }\in$ with $\Re s,\Re s^{\prime }$ sufficiently large, the regularized Siegel–Weil/doubling method yields the Rallis inner product formula (see [56,58,60]):

Theorem 4 

(Rallis inner product formula). With Tamagawa measures and spherical local data, one has

$${〈{\Theta }_{\phi }E(·,s),{\Theta }_{\phi }E(·,s^{\prime })〉}_{\mathrm{reg}}=C_{\mathrm{glob}}{\displaystyle \frac{\zeta (s+s^{\prime })}{\zeta \left(2s\right)\zeta \left(2s^{\prime }\right)}}\xi \left(s\right)\xi \left(s^{\prime }\right),$$

(10)

as a meromorphic identity in $(s,s^{\prime }){\in }^2$, where $C_{\mathrm{glob}}\ne 0$ is an absolute constant independent of $s,s^{\prime }$.

Remark 4. 

All subsequent considerations involve $\partial /\partial \sigma$ of logarithms, so the constant $C_{\mathrm{glob}}$ is irrelevant for the identities derived.

Specializing to $s^{\prime }=1-\overline{s}$ and using $\xi (1-\overline{s})=\overline{\xi \left(s\right)}$ yields

$${\mathcal{J}}_{\Theta }\left(s\right)={〈\Theta (·)E(·,s),\Theta (·)E(·,1-\overline{s})〉}_{\mathrm{reg}}=C_{\mathrm{glob}}{\displaystyle \frac{\zeta (1+2\mathrm{i}t)}{\zeta \left(2s\right)\zeta (2-2\overline{s})}}{\left|\xi \left(s\right)\right|}^2.$$

(11)

Proof 

(Proof of 1). Taking absolute values in (11) gives

$$|{\mathcal{J}}_{\Theta }\left(s\right)|=|C_{\mathrm{glob}}|{\displaystyle \frac{\left|\zeta \right(1+2\mathrm{i}t\left)\right|}{\left|\zeta \left(2s\right)\right||\zeta (2-2\overline{s})|}}{\left|\xi \left(s\right)\right|}^2.$$

Taking real logarithms and differentiating in $\sigma$ yields

$${\displaystyle \frac{\partial }{\partial \sigma }}log\left|{\mathcal{J}}_{\Theta }\left(s\right)\right|=-2\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }\left(2s\right)}{\zeta \left(2s\right)}}+2\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }(2-2\overline{s})}{\zeta (2-2\overline{s})}}+2\mathrm{Re}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}},$$

since both $|C_{\mathrm{glob}}|$ and $\zeta (1+2\mathrm{i}t)$ are independent of $\sigma$. Writing $2s=1+2\sigma +2\mathrm{i}t$ and $2-2\overline{s}=1-2\sigma +2\mathrm{i}t$ yields (5). The interpretation as an identity of tempered distributions follows from 2.    □

Remark 5. 

The factor $\zeta (s+s^{\prime })/\left(\zeta \left(2s\right)\zeta \left(2s^{\prime }\right)\right)$ in (10) is intrinsic to the doubling construction. After specializing $s^{\prime }=1-\overline{s}$, the numerator becomes $\zeta (1+2\mathrm{i}t)$, independent of σ, whereas the denominators $\zeta \left(2s\right)$ and $\zeta (2-2\overline{s})$ retain nontrivial σ–dependence and produce the correction terms in (5). These Euler corrections cancel only on the critical line.

Proof 

(Proof of 2). Convolution with $K_H$ is continuous on ${\mathcal{S}}^{\prime }\left(\mathbb{R}\right)$ and commutes with ${\partial }_{\sigma }$. Integrating (5) against $K_H(t-t_0)$ in t and evaluating at $\sigma ={\sigma }^{★}$ gives the asserted identity.    □

5. Strip Poisson Calculus and Short–Band Freezing

Let $h\in (0,1/2]$ be fixed. For the purposes of this section, we regard $u(\sigma ,t)=log|\xi (\frac{1}{2}+\sigma +\mathrm{i}t)|$ as a tempered distribution in t for each fixed $\sigma$, as justified by 2. Away from zeros and the pole of $\xi$, u is a genuine harmonic function of $(\sigma ,t)$ on vertical strips.

5.1. Strip Poisson Multipliers

Lemma 1 

(Strip Poisson multipliers for the normal derivative). Let $u(\sigma ,t)$ be harmonic on $\left|\sigma \right|\le h$. For any ${\sigma }^{★}\in (-h,h)$ and $\lambda \in \mathbb{R}$, with $F_{\pm }\left(t\right)={\partial }_{\sigma }u(\pm h,t)$ and

$$F_{\mathrm{ev}}={\displaystyle \frac{F_{+}+F_{-}}{2}},F_{\mathrm{odd}}={\displaystyle \frac{F_{+}-F_{-}}{2}},$$

one has

$$\widehat{{\partial }_{\sigma }u}({\sigma }^{★},\lambda )={\tilde{m}}_{\mathrm{ev}}(\lambda ;{\sigma }^{★})\widehat{F_{\mathrm{ev}}}\left(\lambda \right)+{\tilde{m}}_{\mathrm{odd}}(\lambda ;{\sigma }^{★})\widehat{F_{\mathrm{odd}}}\left(\lambda \right),$$

where

$${\tilde{m}}_{\mathrm{ev}}(\lambda ;{\sigma }^{★})={\displaystyle \frac{cosh\left(\right|\lambda \left|{\sigma }^{★}\right)}{cosh\left(\right|\lambda \left|h\right)}},{\tilde{m}}_{\mathrm{odd}}(\lambda ;{\sigma }^{★})={\displaystyle \frac{sinh\left(\right|\lambda \left|{\sigma }^{★}\right)}{sinh\left(\right|\lambda \left|h\right)}}.$$

In particular, as $\left|\lambda \right|\to 0$,

$${\tilde{m}}_{\mathrm{ev}}(\lambda ;{\sigma }^{★})\to 1,{\tilde{m}}_{\mathrm{odd}}(\lambda ;{\sigma }^{★})\to {\displaystyle \frac{{\sigma }^{★}}{h}}.$$

Moreover, for each fixed ${\sigma }^{★}\in (-h,h)$,

$$\left|{\partial }_{\lambda }{\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}(\lambda ;{\sigma }^{★})\right|\ll e^{-(h-|{\sigma }^{★}\left|\right)\left|\lambda \right|}(1+|\lambda \left|\right),\lambda \in \mathbb{R},$$

with an implied constant depending only on h.

Proof. 

Fourier transforming in t gives $({\partial }_{\sigma }^2-{\lambda }^2)\widehat{u}=0$, hence $\widehat{u}(\sigma ,\lambda )=A{e}^{\left|\lambda \right|\sigma }+B{e}^{-\left|\lambda \right|\sigma }$. Matching the normal derivatives at $\sigma =\pm h$ and solving for $A,B$ produces the stated formula. The bounds on ${\partial }_{\lambda }{\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}$ follow by differentiating these explicit expressions and using $cosh\left(\right|\lambda \left|h\right),sinh\left(\right|\lambda \left|h\right)\asymp e^{\left|\lambda \right|h}$ as $\left|\lambda \right|\to \infty$.    □

5.2. Short–Band Freezing

Lemma 2 

(Short–band freezing). Let $B_{{\lambda }_0,\Delta }$ be the band $\left\{\right|\lambda \pm {\lambda }_0|\le 2\Delta \}$ with ${\lambda }_0\asymp T$ and $\Delta =T^{-1-\epsilon }$, and let $P_{\Delta }$ be the corresponding spectral projector. Then, uniformly for ${\sigma }^{★}$ in compact subsets of $(-h,h)$,

$$\underset{\lambda \in B_{{\lambda }_0,\Delta }}{sup}|{\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}(\lambda ;{\sigma }^{★})-{\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}({\lambda }_0;{\sigma }^{★})|\ll \Delta ,$$

and for any $f\in L^2\left(\mathbb{R}\right)$ and $t_0\in \mathbb{R}$,

$$\left|\left({\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}(D_t;{\sigma }^{★})-{\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}({\lambda }_0;{\sigma }^{★})\right)(K_H*P_{\Delta }f)\left(t_0\right)\right|\ll {\Delta }^{3/2}{\parallel f\parallel }_2=o\left(H^{-1}\right){\parallel f\parallel }_2.$$

Here $D_t$ denotes the Fourier multiplier with symbol λ, and the implied constants may depend on h but are uniform for $|{\lambda }_0|\asymp T$.

Proof. 

The mean value theorem and 1 give

$$|{\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}(\lambda ;{\sigma }^{★})-{\tilde{m}}_{\mathrm{ev}/\mathrm{odd}}({\lambda }_0;{\sigma }^{★})|\le |\lambda -{\lambda }_0|\underset{\xi \in B_{{\lambda }_0,\Delta }}{sup}\left|{\partial }_{\lambda }\tilde{m}(\xi ;{\sigma }^{★})\right|\ll \Delta .$$

The corresponding operator on $L^2$ is a multiplier of size $\ll \Delta$ supported on a set of width $\ll \Delta$, so its $L^2\to L^2$ norm is $\ll \Delta$, and Bernstein’s inequality for band–limited functions implies the pointwise bound $\ll {\Delta }^{3/2}{\parallel f\parallel }_2$. With $\Delta =T^{-1-\epsilon }$ and $H\asymp T$, this is $o\left(H^{-1}\right)$ as claimed.    □

5.3. Local Renormalization for ${\partial }_{\sigma }u$

Lemma 3 

(Local renormalization for ${\partial }_{\sigma }u$). Fix $h\in (0,1/2]$, ${\sigma }_0\in (0,h)$ and $\epsilon >0$. For each $T\ge 1$, $H\asymp T$, $|t_0|\le CH$, let ${\mathcal{Z}}_{t_0,H}$ be the multiset consisting of all zeros and the pole ρ of ξ, counted with multiplicities $m_{\rho }\in \mathbb{Z}$, such that $|\Im \rho -t_0|\le 2H$ and $|\Re \rho -\frac{1}{2}|\le h$. Define

$$R_{t_0}^{\left(1\right)}(\sigma ,t)=\sum _{\rho \in {\mathcal{Z}}_{t_0,H}}{m}_{\rho }\mathrm{Re}\left({\displaystyle \frac{1}{\frac{1}{2}+\sigma +\mathrm{i}t-\rho }}\right).$$

Then there exists a harmonic function $\tilde{u}(\sigma ,t)$ on the rectangle $\left\{\right|\sigma |\le h,|t-t_0|\le 2H\}$ such that

$$\tilde{v}(\sigma ,t):={\partial }_{\sigma }\tilde{u}(\sigma ,t)={\partial }_{\sigma }u(\sigma ,t)-R_{t_0}^{\left(1\right)}(\sigma ,t)$$

is harmonic on this rectangle. Moreover, for any short band $P_{\Delta }$ with $\Delta =T^{-1-\epsilon }$ and any ${\sigma }^{★}$ with $|{\sigma }^{★}|\ge {\sigma }_0$,

$${\int }_{\mathbb{R}}{K}_H(t-t_0)\left(P_{\Delta }{R}_{t_0}^{\left(1\right)}\right)({\sigma }^{★},t)dt=O\left(H^{-\eta }\right),$$

uniformly in $|t_0|\le CH$, where $\eta >0$ depends only on ε.

Proof. 

Local analysis of $u\left(s\right)=\mathrm{Re}log\xi \left(s\right)$ near zeros and the pole shows that in a neighbourhood of each such $\rho$ one has

$$u\left(s\right)=m_{\rho }log|s-\rho |+h_{\rho }\left(s\right),$$

with $h_{\rho }$ harmonic. Subtracting these logarithmic terms over $\rho \in {\mathcal{Z}}_{t_0,H}$ yields

$$\tilde{u}(\sigma ,t):=u(\sigma ,t)-\sum _{\rho \in {\mathcal{Z}}_{t_0,H}}{m}_{\rho }log\left|\frac{1}{2}+\sigma +\mathrm{i}t-\rho \right|,$$

which is harmonic on $\left\{\right|\sigma |\le h,|t-t_0|\le 2H\}$ and satisfies ${\partial }_{\sigma }\tilde{u}={\partial }_{\sigma }u-R_{t_0}^{\left(1\right)}$.

For the bound on the renormalization term, fix ${\sigma }^{★}$ with $|{\sigma }^{★}|\ge {\sigma }_0$ and write each summand of $R_{t_0}^{\left(1\right)}({\sigma }^{★},t)$ as

$$f_{\delta ,\gamma }\left(t\right)=\mathrm{Re}\left({\displaystyle \frac{1}{\delta +\mathrm{i}(t-\gamma )}}\right)={\displaystyle \frac{\delta }{{\delta }^2+{(t-\gamma )}^2}},\delta ={\sigma }^{★}-(\Re \rho -\frac{1}{2}),\gamma =\Im \rho .$$

Its Fourier transform is explicit:

$${\widehat{f}}_{\delta ,\gamma }\left(\lambda \right)=\pi \mathrm{sgn}\left(\delta \right){\mathrm{e}}^{-\mathrm{i}\lambda \gamma }{\mathrm{e}}^{-\left|\delta \right|\left|\lambda \right|},\lambda \in \mathbb{R},$$

so in particular

$$\left|{\widehat{f}}_{\delta ,\gamma }\left(\lambda \right)\right|\le \pi {\mathrm{e}}^{-\left|\delta \right|\left|\lambda \right|}.$$

See, for example, [53]. Therefore

$$\widehat{K_H*P_{\Delta }{f}_{\delta ,\gamma }}\left(\lambda \right)={\widehat{K}}_H\left(\lambda \right){\mathbf{1}}_{B_{{\lambda }_0,\Delta }}\left(\lambda \right){\widehat{f}}_{\delta ,\gamma }\left(\lambda \right),$$

and so

$$\parallel K_H*P_{\Delta }{f}_{\delta ,\gamma }{\parallel }_{L^{\infty }}\ll {\int }_{B_{{\lambda }_0,\Delta }}{\mathrm{e}}^{-\left|\delta \right|\left|\lambda \right|}d\lambda \ll \Delta ,$$

uniformly in $\delta$ and $\gamma$. The number of $\rho$ with $|\Im \rho -t_0|\le 2H$ and $|\Re \rho -\frac{1}{2}|\le h$ is $O(HlogT)$ by the zero–counting bound $N\left(T\right)\ll TlogT$ (see, for instance, [14]). Hence

$${\int }_{\mathbb{R}}{K}_H(t-t_0)\left(P_{\Delta }{R}_{t_0}^{\left(1\right)}\right)({\sigma }^{★},t)dt\ll H\Delta logT\asymp T^{-\epsilon }logT=O\left(H^{-\eta }\right)$$

for some $\eta >0$ depending only on $\epsilon$.    □

Remark 6 

(Local harmonicity vs. global subharmonicity). For $u(\sigma ,t)=log|\xi (\frac{1}{2}+\sigma +\mathrm{i}t)|$, the function u is subharmonic on the strip $\left|\sigma \right|\le h$ and harmonic away from the zeros and the pole of ξ. Lemma 3 constructs, for each window centered at $t_0$, a locally harmonic function $\tilde{u}$ on the rectangle $\left\{\right|\sigma |\le h,|t-t_0|\le 2H\}$ by subtracting the logarithmic singularities arising from zeros and the pole in that window. It is this locally harmonic function $\tilde{u}$ to which the strip Poisson calculus of 1 is applied in the proof of 3.

Remark 7 

(Short Fourier–side proof note). The key ingredients are: (i) the explicit transform $\widehat{\delta /({\delta }^2+{(t-\gamma )}^2)}\left(\lambda \right)\asymp {\mathrm{e}}^{-\left|\delta \right|\left|\lambda \right|}$, which exhibits exponential damping in frequency proportional to the horizontal distance $\left|\delta \right|$ from the zero (or pole); and (ii) the standard zero–counting estimate $N\left(T\right)\ll TlogT$, which bounds the number of such singularities in a given height window. The band width is $\asymp \Delta$, and the Fejér multiplier is bounded on $B_{{\lambda }_0,\Delta }$, leading to the $O(H\Delta logT)$ bound.

5.4. Proof of the Strip Bridge

Proof 

(Proof of 3). Write ${\partial }_{\sigma }u=\tilde{v}+R_{t_0}^{\left(1\right)}$ as in 3, and let $\tilde{u}$ be the harmonic function constructed there so that $\tilde{v}={\partial }_{\sigma }\tilde{u}$. Applying 1 to $\tilde{u}$ gives, in frequency,

$$\widehat{\tilde{v}}({\sigma }^{★},\lambda )={\tilde{m}}_{\mathrm{ev}}(\lambda ;{\sigma }^{★})\widehat{{\tilde{F}}_{\mathrm{ev}}}\left(\lambda \right)+{\tilde{m}}_{\mathrm{odd}}(\lambda ;{\sigma }^{★})\widehat{{\tilde{F}}_{\mathrm{odd}}}\left(\lambda \right),$$

where ${\tilde{F}}_{\pm }\left(t\right)={\partial }_{\sigma }\tilde{u}(\pm h,t)=F_{\pm }\left(t\right)-R_{t_0}^{\left(1\right)}(\pm h,t)$, and ${\tilde{F}}_{\mathrm{ev}/\mathrm{odd}}$ are defined analogously.

Because $P_{\Delta }$ and convolution by $K_H$ commute with Fourier multipliers, and $B_{{\lambda }_0,\Delta }\subset [-H,H]$,

$$K_H*P_{\Delta }\left({\partial }_{\sigma }u\right)({\sigma }^{★},·)={\tilde{m}}_{\mathrm{ev}}(D_t;{\sigma }^{★})K_H*P_{\Delta }{F}_{\mathrm{ev}}+{\tilde{m}}_{\mathrm{odd}}(D_t;{\sigma }^{★})K_H*P_{\Delta }{F}_{\mathrm{odd}}+E,$$

where the error E collects the $K_H*P_{\Delta }$ transforms of $R_{t_0}^{\left(1\right)}$ and of $R_{t_0}^{\left(1\right)}(\pm h,·)$ fed through the multipliers. By 3 and the same Fourier–side argument for the boundary terms, each such contribution is $O\left(H^{-\eta }\right)$ uniformly for $|{\sigma }^{★}|\ge {\sigma }_0$.

Finally, 2 is applied to freeze the multipliers at ${\lambda }_0$, introducing an $o\left(H^{-1}\right)$ error that can be absorbed into the $O\left(H^{-\eta }\right)$ term (possibly with a smaller $\eta >0$). Evaluating at $t_0$ gives (7).    □

6. Fejér Smoothing, Cusp Truncation with Counterterms, and Stability

In this section the regularized inner product is compared to a counterterm–subtracted truncated integral. The correct object for comparison with ${\langle ·,·\rangle }_{\mathrm{reg}}$ is the truncated integral of the original integrand with the Zagier–Arthur cusp counterterms removed.

Let $\mathcal{F}$ be a fixed fundamental domain for X, and for $Y\ge 1$ let ${\mathcal{F}}_Y$ be its standard truncation at height Y (cf. [15,64]). Define

$$\mathsf{I}(s;Y):={\int }_{{\mathcal{F}}_Y}\Theta \left(z\right)E(z,s)\overline{\Theta \left(z\right)E(z,1-\overline{s})}d\mu \left(z\right).$$

The cusp asymptotics (9) and (1) show that the constant term of $\Theta \left(z\right)E(z,s)\overline{\Theta \left(z\right)E(z,1-\overline{s})}$ in the cusp consists of a finite linear combination of monomials in y and $logy$ with s–dependent coefficients. Integrating termwise in y over $[1,Y]$ and in x over $[-1/2,1/2]$ produces finitely many explicit counterterms ${\left\{{\mathsf{CT}}_j(s;Y)\right\}}_{j=1}^J$, each a polynomial in Y and $logY$ with coefficients depending meromorphically on s, such that (cf. [64])

$${\mathcal{J}}_{\Theta }\left(s\right)=\underset{Y\to \infty }{lim}\left(\mathsf{I}(s;Y)-\sum _{j=1}^J{\mathsf{CT}}_j(s;Y)\right)$$

(12)

and, moreover, the tail

$$\mathsf{R}(s;Y):=\left(\mathsf{I}(s;Y)-\sum _{j=1}^J{\mathsf{CT}}_j(s;Y)\right)-{\mathcal{J}}_{\Theta }\left(s\right)$$

satisfies

$$\forall M\ge 0,\mathsf{R}(s;Y)=O_M\left(Y^{-M}\right),$$

(13)

locally uniformly on vertical strips in s. This follows from the fact that, after removal of the finitely many nondecaying terms in the cusp, the remaining integrand decays exponentially in y and its integral over $y\ge Y$ is $O\left(e^{-cY}\right)$.

Set

$${\mathsf{I}}_{\mathrm{reg}}(s;Y):=\mathsf{I}(s;Y)-\sum _{j=1}^J{\mathsf{CT}}_j(s;Y).$$

Proposition 1 

(Truncation stability under Fejér smoothing with counterterms). Fix ${\sigma }^{★}$ in a compact subset of $\mathbb{R}$ and $t_0\in \mathbb{R}$. For every $A>0$ there exists $B=B\left(A\right)>0$ such that, setting $Y=H^B$,

$$\begin{array}{cc}\hfill & {\int }_{\mathbb{R}}{K}_H(t-t_0){\displaystyle \frac{\partial }{\partial \sigma }}log\left|{\mathsf{I}}_{\mathrm{reg}}\left(\frac{1}{2}+\sigma +\mathrm{i}t;Y\right)\right|dt{|}_{\sigma ={\sigma }^{★}}\hfill \\ \hfill & ={\int }_{\mathbb{R}}{K}_H(t-t_0){\displaystyle \frac{\partial }{\partial \sigma }}log\left|{\mathcal{J}}_{\Theta }\left(\frac{1}{2}+\sigma +\mathrm{i}t\right)\right|dt{|}_{\sigma ={\sigma }^{★}}+O\left(H^{-A}\right),\hfill \end{array}$$

uniformly for $|t_0|\le CH$. The implied constant may depend on A and ${\sigma }^{★}$ but is independent of H and $t_0$.

Proof. 

By (13),

$${\mathsf{I}}_{\mathrm{reg}}(s;Y)={\mathcal{J}}_{\Theta }\left(s\right)(1+\epsilon (s;Y)),\epsilon (s;Y)=O_M\left(Y^{-M}\right),$$

for any chosen $M\ge 1$, locally uniformly on vertical strips. Differentiating in $\sigma$ and using Cauchy estimates on $\epsilon$ in small discs in s implies ${\partial }_{\sigma }\epsilon (s;Y)=O_M\left(Y^{-M}\right)$ as well. Hence

$${\displaystyle \frac{\partial }{\partial \sigma }}log\left|{\mathsf{I}}_{\mathrm{reg}}(s;Y)\right|={\displaystyle \frac{\partial }{\partial \sigma }}log\left|{\mathcal{J}}_{\Theta }\left(s\right)\right|+O_M\left(Y^{-M}\right),$$

pointwise away from zeros and poles of ${\mathcal{J}}_{\Theta }$ and in the sense of distributions in t in general. Convolution in t against $K_H$ preserves the bound, uniformly for $|t_0|\le CH$. Given $A>0$, choose M and then $B>0$ such that $Y^{-M}=H^{-BM}\ll H^{-A}$; for instance $B\left(A\right)=2A/M$ suffices. This yields the claim.    □

Remark 8. 

Without subtracting the Zagier–Arthur counterterms, $\mathsf{I}(s;Y)$ differs from ${\mathcal{J}}_{\Theta }\left(s\right)$ by explicit polynomials in Y and $logY$ with coefficients depending meromorphically on s. These do not decay with Y and would persist after Fejér smoothing. The counterterms remove precisely these divergent pieces, and the remaining cusp tail is exponentially small in Y, which is the source of the $O\left(H^{-A}\right)$ stability in 1.

7. Numerical Verification in a Sample Region

This brief section records a qualitative description of numerical checks supporting the identity of 1. It is included only to indicate that the analytic formula admits direct numerical verification in a modest range; no new insights are claimed.

Description of the Numerical Scheme

Let $s=\frac{1}{2}+\sigma +\mathrm{i}t$ with a modest choice, for example $\sigma =0.1$ and t in the range $5\le t\le 20$. The following steps may be implemented:

(1)

Approximate ${\mathcal{J}}_{\Theta }\left(s\right)$ by truncation. Use the Fourier expansion of $E(z,s)$ and the truncated double sum defining $\theta \left(z\right)$ to approximate $\Theta \left(z\right)$ and $E(z,s)$ on a rectangular grid $(x,y)$ with $\left|x\right|\le 1/2$ and $1\le y\le Y$, for a fixed but large Y (e.g. $Y\in [20,100]$). Integrate $\Theta \left(z\right)E(z,s)\overline{\Theta \left(z\right)E(z,1-\overline{s})}$ over this domain with the hyperbolic measure, and subtract the explicit Zagier–Arthur counterterms described in §Section 6 to obtain a numerical approximation to ${\mathsf{I}}_{\mathrm{reg}}(s;Y)$.

(2)

Approximate ${\partial }_{\sigma }log\left|{\mathcal{J}}_{\Theta }\left(s\right)\right|$ by finite differences. For a small step $h>0$ (for instance $h={10}^{-3}$), form

$$D_h\left(s\right):={\displaystyle \frac{log|{\mathsf{I}}_{\mathrm{reg}}(s+h;Y)|-log|{\mathsf{I}}_{\mathrm{reg}}(s-h;Y)|}{2h}}.$$

Proposition 1 shows that, with Y chosen as a sufficiently large power of H (and H comparable to $\left|t\right|$), the difference between $D_h\left(s\right)$ and ${\partial }_{\sigma }log\left|{\mathcal{J}}_{\Theta }\left(s\right)\right|$ is bounded by a power of $H^{-1}$, plus the usual finite–difference discretization error.

(3)

Evaluate the right–hand side. The right–hand side of (5),

$$R\left(s\right):=2\mathrm{Re}{\displaystyle \frac{{\xi }^{\prime }\left(s\right)}{\xi \left(s\right)}}-2\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }\left(2s\right)}{\zeta \left(2s\right)}}+2\mathrm{Re}{\displaystyle \frac{{\zeta }^{\prime }(2-2\overline{s})}{\zeta (2-2\overline{s})}},$$

can be evaluated using high–precision complex arithmetic and standard routines (or finite differences) for ${\zeta }^{\prime }/\zeta$ and ${\Gamma }^{\prime }/\Gamma$.

Preliminary implementations following this scheme, with reasonable truncation parameters for the Eisenstein series and the theta series and with Y chosen sufficiently large, show that the numerical values of $D_h\left(s\right)$ and $R\left(s\right)$ agree to several digits in the above sample range of s, in line with the analytic error estimates.

Remark 9. 

No attempt is made here to optimize the numerical scheme or to investigate regions of large t. The purpose is only to indicate that (5) can be checked numerically for moderate s directly from the defining integrals, once the counterterms are correctly taken into account.

Appendix B.1. Weil Representation and the Spherical Matrix Coefficient

A direct computation shows that for $z=x+\mathrm{i}y$ and $t\ge 0$ related by $d_{\mathbb{H}}(z,\mathrm{i})=2t$,

$$\langle {\omega }_{\infty }\left(g_z\right){\phi }_{\infty },{\phi }_{\infty }\rangle ={\displaystyle \frac{y^{1/2}}{\sqrt{{(y+1)}^2+x^2}}}={\displaystyle \frac{1}{2cosht}}.$$

Appendix B.2. Cartan Decomposition and the Spherical Section

With Haar measure normalized so that $dg=sinh\left(2t\right)dtdkd{k}^{\prime }$ on $G\left(\mathbb{R}\right)=K{A}^{+}K$, one may take $f_{s,\infty }\left(a_t\right)=c_{\infty }\left(s\right){(2cosht)}^{-s}$.

Appendix B.3. Evaluation of Z ∞ (s,φ ∞ ,f s,∞ )

One obtains

$$Z_{\infty }(s,{\phi }_{\infty },f_{s,\infty })=c_{\infty }\left(s\right){\int }_0^{\infty }{(2cosht)}^{-s}{\displaystyle \frac{1}{2cosht}}sinh\left(2t\right)dt=c_{\infty }\left(s\right){\int }_0^{\infty }{(2cosht)}^{-s}sinhtdt.$$

With substitutions $u=tanht$ and then $v=u^2$,

$${\int }_0^{\infty }{(2cosht)}^{-s}sinhtdt=2^{-s-1}{\displaystyle \frac{\Gamma \left(\frac{s-1}{2}\right)}{\Gamma \left(\frac{s+1}{2}\right)}}.$$

Choosing

$$c_{\infty }\left(s\right)=2^{s+1}{\pi }^{-s/2}\Gamma \left({\displaystyle \frac{s}{2}}\right){\displaystyle \frac{\Gamma \left(\frac{s+1}{2}\right)}{\Gamma \left(\frac{s-1}{2}\right)}}$$

gives $Z_{\infty }(s,{\phi }_{\infty },f_{s,\infty })={\pi }^{-s/2}\Gamma (s/2)$, matching the archimedean $\Gamma$–factor of $\xi \left(s\right)$. Together with the unramified finite place identities $Z_p(s,{\phi }_p,f_{s,p})={\zeta }_p\left(s\right)$, this yields ${\prod }_v{Z}_v(s,{\phi }_v,f_{s,v})=\xi \left(s\right)$.