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Zhi Qi

Publications and source records attributed to Zhi Qi.

At least 19 recordsLinked to original sources
arXiv2026

Luo's Spectral Large Sieve Inequality on Short Intervals

Let $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb {Z}) $ with Hecke eigenvalues $\lambda_j (n)$ and Laplace eigenvalue $1/4+t_j^2$. In this paper, we consider the short-interval variant of the twisted spectral large sieve inequality of Luo for $ \lambda_j (n) n^{it_j} $ on the range $t_j \leqslant T$ and prove that the `Eisenstein--Kloosterman' cancellation discovered by Luo is effective on the interval $ T < t_j \leqslant T + M $ as long as $\sqrt{T} < M \leqslant T$. Moreover, our approach yields an improvement of the large sieve inequality of Luo.

math.NT↗
arXiv2026

Hybrid Weyl Subconvexity over Imaginary Quadratic Fields

On imaginary quadratic fields, we establish the $\mathrm{GL}_2$ Bessel $\delta$-method and prove the hybrid Weyl-type subconvexity bound for $\mathrm{GL}_2 \times \mathrm{GL}_1$ twisted $L$-functions in the Archimedean aspect of the Hecke character on $\mathrm{GL}_1$.

math.NT↗
arXiv2026

A Criterion for Equidistribution along the $\Omega$ Function over Polynomial Sequences with Applications

Let $P (Y_1, ..., Y_d)$ be a certain fixed homogeneous polynomial of integral coefficients. In this paper, we establish a quantitative equidistribution criterion for the ergodic averages along $\Omega (|P (n_1, ..., n_d)|)$. Consequently, by an estimate of Lachand, we prove the following variant of a theorem of Bergelson and Richter: if $P$ is an irreducible binary cubic form and $ (X, T)$ is a uniquely ergodic system with unique invariant measure $\mu$, then for any $x \in X$ and $f \in C(X)$, \begin{equation*} \lim_{N \rightarrow \infty} \frac 1 {N^2} {\mathop{\sum\sum}_{n_1, n_2 \leqslant N}} f \big( T^{ \Omega (|P (n_1, n_2)| ) } x \big) = \int_{X} f \ \mathrm{d} \mu . \end{equation*} Moreover, we prove in the appendix a related conjecture of C\'espedes and Donoso over number fields.

math.DS↗
arXiv2026

On the Second Moment of $ L (1/2, \mathrm{As}(f) \times \phi)$

Let $\mathbf{F} = \mathbf{Q}(\sqrt D)$ be a real quadratic field. In this paper, we establish a large sieve inequality for the Asai lifts $ \mathrm{As} (f) $ with $f$ in a Hecke orthonormal basis of the space of Hilbert modular cusp forms of parallel weight $(k, k)$ over $ \mathbf{F} $. As an application, for a fixed Hecke--Maass cusp form $\phi $ over $\mathbf{Q}$, we prove a non-trivial bound for the second moment of the convoluted central $L$-values $ L (1/2, \mathrm{As}(f) \times \phi) $ in the $k$-aspect.

math.NT↗
arXiv2026

On the Second Moment of $L (1/2, \mathrm{As} (f))$

Let $\mathbf{F}$ be a real quadratic field. Let $f $ traverse a Hecke orthonormal basis of Hilbert cusp forms over $ \mathbf{F} $ of full level and parallel weight $(k,k)$. As $k \rightarrow \infty$, we prove an asymptotic formula for the second moment of central Asai $L$-values $L (1/2, \mathrm{As} (f))$: \begin{equation*} {\sum}_{f } \, \omega_f L(1/2,\mathrm{As}(f))^2 = P_3 ( \log {k } ) k^2 + O_{\mathbf{F},\varepsilon} (k^{3/2 + \varepsilon} ), \end{equation*} where $\omega_f$ are the harmonic weights and $P_3 (X)$ is an explicit polynomial of degree $3$. This refines the mean Lindel\"of bound $ O_{\mathbf{F},\varepsilon} (k^{2 + \varepsilon} ) $ proved by Wenzhi Luo.

math.NT↗
arXiv2025

Secondary Term for the Mean Value of Maass Special $L$-values

In this paper, we discover a secondary term in the asymptotic formula for the mean value of Hecke--Maass special $L$-values $ L (1/2+it_f, f) $ with the average over $f (z)$ in an orthonormal basis of Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). To be explicit, we prove $$ \sum_{t_f \leqslant T} \omega_f L (1/2+it_f, f) = \frac {T^2} {\pi^2} + \frac {4 T^{3/2}} {3\pi^{3/2} } + O \big(T^{1+\varepsilon}\big), $$ for any $\varepsilon > 0$, where $\omega_f$ are the harmonic weights. This provides a new instance of (large) secondary terms in the moments of $L$-functions---it was known previously only for the smoothed cubic moment of quadratic Dirichlet $L$-functions. The proof relies on an explicit formula for the smoothed mean value of $L (1/2+it_f, f)$.

math.NT↗
arXiv2025

The Second Moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ $L$-functions at Special Points

Let $\phi$ be a fixed Hecke--Maass form for $\mathrm{SL}_3 (\mathbb{Z})$ and $u_j $ traverse an orthonormal basis of Hecke--Maass forms for $\mathrm{SL}_2 (\mathbb{Z}) $. Let $1/4+t_j^2$ be the Laplace eigenvalue of $u_j $. In this paper, we prove the mean Lindel\"of hypothesis for the second moment of $ L (1/2+it_j, \phi \times u_j) $ on $ T < t_j \leqslant T + \sqrt{T} $. Previously, this was proven by Young on $ t_j \leqslant T$. Our approach is more direct as we do not apply the Poisson summation formula to detect the `Eisenstein--Kloosterman' cancellation.

math.NT↗
arXiv2025

On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points

In this paper, we consider the non-vanishing problem for the family of special Hecke--Maass $L$-values $ L (1/2+it_f, f) $ with $f (z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4 + t_f^2$ ($t_f > 0$). We prove that 20% of $L (1/2+it_f, f)$ for $ t_f \leqslant T$ do not vanish as $T \rightarrow \infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 20% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on the short interval $|t_f-T| \leqslant T^{\mu}$ for any $0 < \mu < 1$.

math.NT↗
arXiv2025

On the Effective Non-vanishing of Rankin--Selberg $L$-functions at Special Points

Let $Q (z)$ be a holomorphic Hecke cusp newform of square-free level and $u_j (z)$ traverse an orthonormal basis of Hecke--Maass cusp forms of full level. Let $1/4 + t_j^2$ be the Laplace eigenvalue $u_j (z)$. In this paper, we prove that there is a constant $ \gamma (Q) $ expressed as a certain Euler product associated to $Q$ such that at least $ \gamma (Q) / 11 $ of the Rankin--Selberg special $L$-values $L (1/2+it_j, Q \otimes u_j)$ for $ t_j \leqslant T$ do not vanish as $T \rightarrow \infty$. Further, we show that the non-vanishing proportion is at least $\gamma (Q) \cdot (4\mu-3) / (4\mu+7) $ on the short interval $ |t_j - T| \leqslant T^{\mu} $ for any $3/4 < \mu < 1$.

math.NT↗
arXiv2025

Improving Decoupled Posterior Sampling for Inverse Problems using Data Consistency Constraint

Diffusion models have shown strong performances in solving inverse problems through posterior sampling while they suffer from errors during earlier steps. To mitigate this issue, several Decoupled Posterior Sampling methods have been recently proposed. However, the reverse process in these methods ignores measurement information, leading to errors that impede effective optimization in subsequent steps. To solve this problem, we propose Guided Decoupled Posterior Sampling (GDPS) by integrating a data consistency constraint in the reverse process. The constraint performs a smoother transition within the optimization process, facilitating a more effective convergence toward the target distribution. Furthermore, we extend our method to latent diffusion models and Tweedie's formula, demonstrating its scalability. We evaluate GDPS on the FFHQ and ImageNet datasets across various linear and nonlinear tasks under both standard and challenging conditions. Experimental results demonstrate that GDPS achieves state-of-the-art performance, improving accuracy over existing methods.

cs.LG↗
arXiv2025

Expanding-and-Shrinking Binary Neural Networks

While binary neural networks (BNNs) offer significant benefits in terms of speed, memory and energy, they encounter substantial accuracy degradation in challenging tasks compared to their real-valued counterparts. Due to the binarization of weights and activations, the possible values of each entry in the feature maps generated by BNNs are strongly constrained. To tackle this limitation, we propose the expanding-and-shrinking operation, which enhances binary feature maps with negligible increase of computation complexity, thereby strengthening the representation capacity. Extensive experiments conducted on multiple benchmarks reveal that our approach generalizes well across diverse applications ranging from image classification, object detection to generative diffusion model, while also achieving remarkable improvement over various leading binarization algorithms based on different architectures including both CNNs and Transformers.

cs.CV↗
arXiv2024

On the Hankel Transform of Bessel Functions on Complex Numbers and Explicit Spectral Formulae over the Gaussian Field

In this paper, on the complex field $\mathbb{C}$, we prove two integral formulae for the Hankel-Mellin transform and the double Fourier-Mellin transform of Bessel functions, both resulting the hypergeometric function. As two applications, we use the former integral formula to make explicit the spectral formula of Bruggeman and Motohashi for the fourth moment of Dedekind zeta function over the Gaussian number field $\mathbb{Q}(i)$ and to establish a spectral formula for the Hecke-eigenvalue twisted second moment of central $L$-values for the Picard group $\mathrm{PGL}_2 (\mathbb{Z}[i])$. Moreover, we develop the theory of distributional Hankel transform on $\mathbb{C} \smallsetminus \{0\}$.

math.NT↗
arXiv2024

Symmetric Square Large Sieve for $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $ and Prime Geodesic Theorem for $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $

In this paper, we improve the error term in the prime geodesic theorem for the Picard manifold $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $. Instead of $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $, we establish a spectral large sieve inequality for symmetric squares over $\mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $. This enables us to improve the bound $ O (T^{3+2/3+\varepsilon}) $ of Balkanova and Frolenkov to $ O (T^{3+1/2+\varepsilon}) $ for the second moment of symmetric square $L$-functions over $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $. The basic idea is to enlarge the spherical family $\Pi_c^{0} (T)$ of Maass cusp forms on $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathbb{H}^3 $ to the family $ \Pi_c (T, \sqrt{T}) $ of cuspidal representations on $ \mathrm{PSL}_2 (\mathbb{Z} {[i]}) \backslash \mathrm{PSL}_2 (\mathbb{C}) $.

math.NT↗
arXiv2024

Hybrid Weyl-type bound for $p$-power twisted $\mathrm{GL} (2)$ $L$-functions

Let $g$ be a fixed holomorphic cusp form of arbitrary level and nebentypus. Let $χ$ be a primitive character of prime-power modulus $q = p^γ$. In this paper, we prove the following hybrid Weyl-type subconvexity bound \begin{align*} L (1/2 + it, g \otimes χ) \ll_{g, p, \varepsilon} ( (1+|t|) q )^{1/3+ \varepsilon} \end{align*} for any $\varepsilon > 0$.

math.NT↗
arXiv2022

RepBNN: towards a precise Binary Neural Network with Enhanced Feature Map via Repeating

Binary neural network (BNN) is an extreme quantization version of convolutional neural networks (CNNs) with all features and weights mapped to just 1-bit. Although BNN saves a lot of memory and computation demand to make CNN applicable on edge or mobile devices, BNN suffers the drop of network performance due to the reduced representation capability after binarization. In this paper, we propose a new replaceable and easy-to-use convolution module RepConv, which enhances feature maps through replicating input or output along channel dimension by $β$ times without extra cost on the number of parameters and convolutional computation. We also define a set of RepTran rules to use RepConv throughout BNN modules like binary convolution, fully connected layer and batch normalization. Experiments demonstrate that after the RepTran transformation, a set of highly cited BNNs have achieved universally better performance than the original BNN versions. For example, the Top-1 accuracy of Rep-ReCU-ResNet-20, i.e., a RepBconv enhanced ReCU-ResNet-20, reaches 88.97% on CIFAR-10, which is 1.47% higher than that of the original network. And Rep-AdamBNN-ReActNet-A achieves 71.342% Top-1 accuracy on ImageNet, a fresh state-of-the-art result of BNNs. Code and models are available at:https://github.com/imfinethanks/Rep_AdamBNN.

cs.CV↗