# P12923: Exact-Gamma Four-Swap Asymptotics for Long Mellin-Radial Amplifiers in the Gaussian Angular Hecke Family

- ID: `P12923`
- Reference: `exact-gamma-four-swap-mellin-radial-amplifiers-gaussian-hecke`
- Page: https://theoremdb.org/statements/P12923
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/exact-gamma-four-swap-mellin-radial-amplifiers-gaussian-hecke?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(\lambda\) be the Gaussian angular Hecke character on the nonzero integral ideals of \(\mathbb Z[i]\), defined on principal ideals by \(\lambda((z))=(z/|z|)^4\), and put \(L_k(s)=L(s,\lambda^k)\) for \(k\ge 1\). Fix \(\Phi\in C_c^\infty((1,2))\), a real number \(0<\vartheta<1/2\), and set \(X=K^\vartheta\). Fix a compact set \(T\subset\mathbb R\) and a sufficiently large integer \(J\). For each \(K\), let \(F_K\in C_c^\infty((0,1))\) satisfy
\[
\sup_K\sum_{j=0}^{J}\int_{\mathbb R}(1+|\xi|)^J\left|\partial_\xi^j\widehat F_K(\xi)\right|\,d\xi<\infty,
\]
where \(\widehat F_K(\xi)=\int_{\mathbb R}F_K(u)e^{-iu\xi}\,du\). For \(t_0\in T\), define
\[
a_{\mathfrak a}=(N\mathfrak a)^{-1/2-it_0}F_K\!\left(\frac{\log N\mathfrak a}{\log X}\right),\qquad A_k(a)=\sum_{\mathfrak a}a_{\mathfrak a}\lambda(\mathfrak a)^k,
\]
and \(\|a\|_2^2=\sum_{\mathfrak a}|a_{\mathfrak a}|^2\). Define
\[
\mathcal M_K(\alpha,\beta;a)=\sum_{k\ge1}\Phi(k/K)L_k(1/2+\alpha)L_k(1/2+\beta)|A_k(a)|^2.
\]
Put \(L_0(s)=\zeta(s)L(s,\chi_{-4})\) and
\[
Z(\gamma,\delta)=\frac{\zeta(1+2\gamma)\zeta(1+2\delta)L_0(1+\gamma+\delta)}{\zeta(2+2\gamma+2\delta)}.
\]
For integral ideals \(\mathfrak n\), define
\[
\tau_{\gamma,\delta}(\mathfrak n)=\sum_{\mathfrak r\mathfrak s=\mathfrak n}N\mathfrak r^{-\gamma}N\mathfrak s^{-\delta}.
\]
For a nonzero ideal \(\mathfrak l\), write \(\mathfrak l=(q_{\mathfrak l})\mathfrak l^\circ\), where \((q_{\mathfrak l})\) is the largest ideal generated by a positive rational integer dividing \(\mathfrak l\). If \(\mathfrak p^d\Vert\mathfrak l^\circ\), define
\[
P_{\mathfrak p^d}(\gamma,\delta)=\frac{N\mathfrak p^{-d/2}}{1+N\mathfrak p^{-1-\gamma-\delta}}\left(\tau_{\gamma,\delta}(\mathfrak p^d)-N\mathfrak p^{-1-2\gamma-2\delta}\tau_{\gamma,\delta}(\mathfrak p^{d-2})\right),
\]
with \(\tau_{\gamma,\delta}(\mathfrak p^j)=0\) for \(j<0\), and set
\[
P_{\mathfrak l}(\gamma,\delta)=\prod_{\mathfrak p^d\Vert\mathfrak l^\circ}P_{\mathfrak p^d}(\gamma,\delta).
\]
This convention includes the ramified prime above \(2\); rational ideal content does not change the angular twist.

For \(s\in\mathbb C\), define
\[
X_k(s)=\pi^{2s}\frac{\Gamma(2k+\tfrac12-s)}{\Gamma(2k+\tfrac12+s)}
\]
and, for \(\epsilon_1,\epsilon_2\in\{\pm1\}\),
\[
G_{\epsilon_1,\epsilon_2}(k;\alpha,\beta)=X_k(\alpha)^{(1-\epsilon_1)/2}X_k(\beta)^{(1-\epsilon_2)/2}.
\]
Define the exact-gamma four-swap main term by
\[
\begin{aligned}
\mathcal M_K^{(4),\mathrm{exact}}(\alpha,\beta;a)=&\sum_{\epsilon_1,\epsilon_2=\pm1}\sum_{k\ge1}\Phi(k/K)G_{\epsilon_1,\epsilon_2}(k;\alpha,\beta)\\
&\times\sum_{\mathfrak a,\mathfrak b}a_{\mathfrak a}\overline{a_{\mathfrak b}}\,Z(\epsilon_1\alpha,\epsilon_2\beta)P_{\mathfrak a\overline{\mathfrak b}}(\epsilon_1\alpha,\epsilon_2\beta).
\end{aligned}
\]
The combined four-sector expression is interpreted by common meromorphic continuation whenever separated terms have removable singularities. Define
\[
H_k(\alpha,\beta)=\pi^{\alpha+\beta}\frac{\Gamma(2k+\tfrac12)^2}{\Gamma(2k+\tfrac12+\alpha)\Gamma(2k+\tfrac12+\beta)}.
\]
Call a compact set \(\Omega\subset\mathbb C^2\) regular if the combined four-sector meromorphic expression has no nonremovable pole on \(\Omega\). The collision divisors \(\alpha=0\), \(\beta=0\), \(\alpha+\beta=0\), and \(\alpha-\beta=0\) may be included when the combined expression has a removable continuation there.

Prove that for every fixed regular compact \(\Omega\), every \(\varepsilon>0\), and every fixed \(0<\vartheta<1/2\), uniformly for \((\alpha,\beta)\in\Omega\), \(t_0\in T\), and all packets \(F_K\) satisfying the stated norm condition,
\[
\mathcal M_K(\alpha,\beta;a)=\mathcal M_K^{(4),\mathrm{exact}}(\alpha,\beta;a)+O_{\Omega,\vartheta,T,\Phi,J,\varepsilon}\!\left(K^{1/2+\varepsilon}\sup_{k\asymp K}|H_k(\alpha,\beta)|\,\|a\|_2^2\right),
\]
with no hidden factor \(X^\delta\) for any fixed \(\delta>0\). Equivalently, by uniform Stirling asymptotics on every fixed regular compact \(\Omega\),
\[
\mathcal M_K=\mathcal M_K^{(4),\mathrm{exact}}+O_{\Omega,\vartheta,T,\Phi,J,\varepsilon}\!\left(K^{1/2-\Re(\alpha+\beta)+\varepsilon}\|a\|_2^2\right).
\]
The endpoint \(\vartheta=1/2\) and shift ranges growing with \(K\) are excluded.

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

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## References

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