# P12921: Square-Root Four-Swap Asymptotics for Mellin-Radial Amplifiers in the Gaussian Angular Hecke Family

- ID: `P12921`
- Reference: `four-swap-mellin-radial-amplifiers-gaussian-hecke`
- Page: https://theoremdb.org/statements/P12921
- 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/four-swap-mellin-radial-amplifiers-gaussian-hecke?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(K\to\infty\). Let \(\lambda\) be the Gaussian angular Hecke character on the nonzero ideals of \(\mathbb Z[i]\), defined by \(\lambda((z))=(z/|z|)^4\), and let \(L_k(s)=L(s,\lambda^k)\). Fix a smooth weight \(\Phi\) supported in \((1,2)\), a real number \(\vartheta\) with \(0<\vartheta<1/2\), and put \(X=K^\vartheta\). Fix also a compact set \(T\subset\mathbb R\), a compact regular set \(\Omega\) of pairs of shifts \((\alpha,\beta)\), and a sufficiently large fixed integer \(J\). For every \(K\), let \(F_K\in C_c^\infty((0,1))\) satisfy
\[
\int_{\mathbb R}(1+|\xi|)^J|\widehat F_K(\xi)|\,d\xi\le 1.
\]
For \(t_0\in T\), define
\[
a_{\mathfrak a}=(N\mathfrak a)^{-1/2-it_0}F_K\!\left(\frac{\log N\mathfrak a}{\log X}\right),\qquad
A_k=\sum_{N\mathfrak a\le X}a_{\mathfrak a}\lambda(\mathfrak a)^k,
\]
where the sums are over nonzero integral ideals of \(\mathbb Z[i]\). Consider the amplified shifted second moment
\[
\mathcal M_K(\alpha,\beta;a)=\sum_{k\ge1}\Phi(k/K)|A_k|^2L_k(1/2+\alpha)L_k(1/2+\beta).
\]
For a nonzero ideal \(\mathfrak l\), define its fixed-twist four-swap main term as follows. Let \(\widetilde\Phi(s)=\int_1^2x^s\Phi(x)\,dx\), let \(L_0(s)=\zeta(s)L(s,\chi_{-4})\), and put
\[
Z(\gamma,\delta)=\frac{\zeta(1+2\gamma)\zeta(1+2\delta)L_0(1+\gamma+\delta)}{\zeta(2+2\gamma+2\delta)}.
\]
Write \(\mathfrak l=(q_{\mathfrak l})\mathfrak l^\circ\), where \((q_{\mathfrak l})\) is the largest ideal generated by a positive rational integer that divides \(\mathfrak l\). If \(\mathfrak p^d\Vert\mathfrak l^\circ\), define
\[
\tau_{\gamma,\delta}(\mathfrak n)=\sum_{\mathfrak r\mathfrak s=\mathfrak n}N\mathfrak r^{-\gamma}N\mathfrak s^{-\delta},
\]
\[
P_{\mathfrak p^d}(\gamma,\delta)=\frac{N\mathfrak p^{-d/2}}{1+N\mathfrak p^{-1-\gamma-\delta}}\left(\tau_{\gamma,\delta}(\mathfrak p^d)-N\mathfrak p^{-1-2\gamma-2\delta}\tau_{\gamma,\delta}(\mathfrak p^{d-2})\right),
\]
with \(\tau_{\gamma,\delta}(\mathfrak p^j)=0\) for \(j<0\), and set \(P_{\mathfrak l}(\gamma,\delta)=\prod_{\mathfrak p^d\Vert\mathfrak l^\circ}P_{\mathfrak p^d}(\gamma,\delta)\). Then define
\[
\mathcal M_{\mathfrak l}^{(4)}(\alpha,\beta)=K\sum_{\epsilon_1,\epsilon_2\in\{\pm1\}}\left(\frac{2K}{\pi}\right)^{-(1-\epsilon_1)\alpha-(1-\epsilon_2)\beta}\widetilde\Phi\bigl(-(1-\epsilon_1)\alpha-(1-\epsilon_2)\beta\bigr)Z(\epsilon_1\alpha,\epsilon_2\beta)P_{\mathfrak l}(\epsilon_1\alpha,\epsilon_2\beta),
\]
with the combined expression interpreted by meromorphic continuation when separated terms have removable singularities. Finally put
\[
\mathcal M_K^{(4)}(\alpha,\beta;a)=\sum_{\mathfrak a,\mathfrak b}a_{\mathfrak a}\overline{a_{\mathfrak b}}\,\mathcal M_{\mathfrak a\overline{\mathfrak b}}^{(4)}(\alpha,\beta).
\]
Prove that for every \(\varepsilon>0\), uniformly for \((\alpha,\beta)\in\Omega\), \(t_0\in T\), and every packet \(F_K\) satisfying the stated norm bound,
\[
\mathcal M_K(\alpha,\beta;a)=\mathcal M_K^{(4)}(\alpha,\beta;a)+O_{\vartheta,\Omega,T,\Phi,J,\varepsilon}\bigl(K^{1/2+\varepsilon}\|a\|_2^2\bigr).
\]
The remainder is required to have no hidden factor \(X^\delta\) with fixed \(\delta>0\). The endpoint \(\vartheta=1/2\) and shift ranges growing with \(K\) are not part of the base problem.

## Status

**Resolution status:** Open.

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

## Research packet

### Working on this

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## References

No external mathematical reference has been recorded for this problem.