arXiv subjects

Nathan Ng

Publications and source records attributed to Nathan Ng.

At least 19 recordsLinked to original sources

arXiv2026

Bounds for Mertens Sums

In this article we provide new bounds for the Mertens sums and products including $\sum_{p \le x} p^{-1}$ and $\prod_{p \le x} (1-\frac{1}{p})$ which provide superior exponential and log type bounds for these sums in all ranges. These weighted prime number sums and products were extensively studied by Rosser and Schoenfeld (1962) and are employed in a wide range of applications in number theory, cryptography, and combinatorics. Extensive tables are provided in this article which will be useful for these types of applications. The main new ideas in this article are sharp bounds for weighted sums of zeros of zeros of the zeta function. We make use of a novel technique of Fiori-Kadiri-Swidinsky (2023) which relies on a recent explicit zero-density estimate for $N(\sigma,T)$ of Kadiri-Lumley-Ng (2018). The bounds and techniques in this article for weighted zeros sums will likely be useful in many other arithmetic applications. Our main theorem significantly improves the exponential decay result of Vanlalngaia (2017) and fills a gap in the literature by correcting work of Dusart (2018). The results are also presented in a way that are amenable to future improvements. In addition, we prove an exact ``Riemann-Guinand explicit formula" for the Mertens sum $\sum_{p \le x} p^{-1}$ that appears to be new.

math.NT↗

arXiv2026

Collaborative Processing for Multi-Tenant Inference on Memory-Constrained Edge TPUs

IoT applications increasingly rely on on-device AI accelerators to ensure high performance, especially in low-connectivity and safety-critical scenarios. However, the limited on-chip memory of these accelerators forces inference runtimes to swap model segments between host and accelerator memory, incurring significant swapping overheads. While collaborative processing by partitioning model execution across CPU and accelerator resources can reduce accelerator memory pressure and execution overhead, naive partitioning may worsen end-to-end latency by either shifting excessive computation to the CPU or failing to sufficiently reduce swapping, a problem that is further exacerbated in multi-tenant and dynamic environments. To address these issues, we present SwapLess, a system for adaptive, multi-tenant TPU-CPU collaborative inference on memory-constrained Edge TPUs. SwapLess utilizes an analytic queueing model that captures partition-dependent CPU/TPU service times as well as inter- and intra-model swapping overheads across different workload mixes and request rates. Using this model, SwapLess continuously adjusts both the partition point and CPU core allocation online to minimize end-to-end response time with low decision overhead. An implementation on Edge TPU-equipped platforms demonstrates that SwapLess reduces mean latency by up to 63.8% for single-tenant workloads and up to 77.4% for multi-tenant workloads relative to the default Edge TPU compiler.

cs.DC↗

arXiv2025

An effective version of Chebotarev's density theorem

Chebotarev's density theorem asserts that the prime ideals are equidistributed among the conjugacy classes of the Galois group of any normal extension of number fields. An effective version of this theorem was first established by Lagarias and Odlyzko in 1977. In this article, we present an explicit refinement of their statement that applies to all non-rational fields, with every implicit constant expressed explicitly in terms of the field invariants. Additionally, we provide a sharper bound for extensions of sufficiently small degree. Our approach begins by proving an explicit formula for a smoothed prime ideal counting function. This relies on recent zero-free regions for Dedekind zeta functions, improved estimates on the number of low-lying zeros, and precise bounds for sums over the non-trivial zeros of the Dedekind $\zeta$-function.

math.NT↗

arXiv2025

Beyond mean-field dynamics of the Dicke model with non-Markovian dephasing

We present a density matrix-based time dependent projection operator formalism to calculate the beyond mean-field dynamics of systems with non-Markovian local baths and one-to-all interactions. Such models encapsulate the physics of condensed phase systems immersed in optical cavities. We use this method, combined with tensor network influence functionals, to study the dynamics of the Dicke model coupled to non-Markovian local dephasing baths at zero temperature, which has a superradiant phase transition in the mean-field limit. The method corrects a spurious initial state dependence found in the mean-field dynamics and describes the emergence of new time scales which are absent in the mean-field dynamics. Our formalism, based on density matrices, is applicable to other quantum optical systems with one-to-all interactions at finite temperatures.

quant-ph↗

arXiv2025

Prime Number Error Terms

In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In 2012 the author made an analogous conjecture for the true order of the sum of the M\"{o}bius function, $M(x)$. This refined an earlier conjecture of Gonek from the 1990's. In this article we speculate on the true size of a large class of prime number error terms and present a general conjecture. This general conjecture includes both Montgomery's conjecture and the conjecture for $M(x)$ as special cases. Recently, Lamzouri (Springer volume: Essays in Analytic Number Theory, In Honor of Helmut Maier's 70th birthday) showed that an effective linear independence conjecture (ELI) for the zeros of the zeta function implies one of the inequalities in Montgomery's conjecture. In this article we adapt Lamzouri's method to show that a generalized effective linear independence (GELI) conjecture implies a lower bound for general prime number error terms. Furthermore, of independent interest, we prove an $L^2$ bound for almost periodic functions. This allows us to weaken significantly one of the conditions in Lamzouri's main result and also give an improvement of the main theorem in an article of Akbary-Ng-Shahabi (Q. J. Math. 65 (2014), no. 3).

math.NT↗

arXiv2025

To Offload or Not To Offload: Model-driven Comparison of Edge-native and On-device Processing In the Era of Accelerators

Computational offloading is a promising approach for overcoming resource constraints on client devices by moving some or all of an application's computations to remote servers. With the advent of specialized hardware accelerators, client devices can now perform fast local processing of specific tasks, such as machine learning inference, reducing the need for offloading computations. However, edge servers with accelerators also offer faster processing for offloaded tasks than was previously possible. In this paper, we present an analytic and experimental comparison of on-device processing and edge offloading for a range of accelerator, network, multi-tenant, and application workload scenarios, with the goal of understanding when to use local on-device processing and when to offload computations. We present models that leverage analytical queuing results to derive explainable closed-form equations for the expected end-to-end latencies of both strategies, which yield precise, quantitative performance crossover predictions that guide adaptive offloading. We experimentally validate our models across a range of scenarios and show that they achieve a mean absolute percentage error of 2.2% compared to observed latencies. We further use our models to develop a resource manager for adaptive offloading and show its effectiveness under variable network conditions and dynamic multi-tenant edge settings.

cs.DC↗

arXiv2024

Comparative Prime Number Theory Problem List

This is a list of problems that were collected from participants at the Comparative Prime Number Theory Symposium held at UBC from June 17 to June 21, 2024. Its goal is to stimulate research and future collaborations in this growing field. This event was part of the PIMS (Pacific Institute of Mathematical Sciences) Collaborative Research Group L-functions in Analytic Number Theory: 2022- 2025.

math.NT↗

arXiv2024

Correlation Functions From Tensor Network Influence Functionals: The Case of the Spin-Boson Model

We investigate the application of matrix product state (MPS) representations of the influence functionals (IF) for the calculation of real-time equilibrium correlation functions in open quantum systems. Focusing specifically on the unbiased spin-boson model, we explore the use of IF-MPSs for complex time propagation, as well as IF-MPSs for constructing correlation functions in the steady state. We examine three different IF approaches: one based on the Kadanoff-Baym contour targeting correlation functions at all times, one based on a complex contour targeting the correlation function at a single time, and a steady state formulation which avoids imaginary or complex times, while providing access to correlation functions at all times. We show that within the IF language, the steady state formulation provides a powerful approach to evaluate equilibrium correlation functions.

physics.chem-ph↗

arXiv2024

Measuring Stochastic Data Complexity with Boltzmann Influence Functions

Estimating the uncertainty of a model's prediction on a test point is a crucial part of ensuring reliability and calibration under distribution shifts. A minimum description length approach to this problem uses the predictive normalized maximum likelihood (pNML) distribution, which considers every possible label for a data point, and decreases confidence in a prediction if other labels are also consistent with the model and training data. In this work we propose IF-COMP, a scalable and efficient approximation of the pNML distribution that linearizes the model with a temperature-scaled Boltzmann influence function. IF-COMP can be used to produce well-calibrated predictions on test points as well as measure complexity in both labelled and unlabelled settings. We experimentally validate IF-COMP on uncertainty calibration, mislabel detection, and OOD detection tasks, where it consistently matches or beats strong baseline methods.

cs.LG↗

arXiv2024

Moments of $L$-functions Problem List

This is an ongoing list of problems that has resulted from the PIMS (Pacific Institute of Mathematical Sciences) Collaborative Research Group L-functions in Analytic Number Theory: 2022- 2025. The focus of this list is on Moments of $L$-functions and related topics.

math.NT↗

arXiv2024

Improving Black-box Robustness with In-Context Rewriting

Machine learning models for text classification often excel on in-distribution (ID) data but struggle with unseen out-of-distribution (OOD) inputs. Most techniques for improving OOD robustness are not applicable to settings where the model is effectively a black box, such as when the weights are frozen, retraining is costly, or the model is leveraged via an API. Test-time augmentation (TTA) is a simple post-hoc technique for improving robustness that sidesteps black-box constraints by aggregating predictions across multiple augmentations of the test input. TTA has seen limited use in NLP due to the challenge of generating effective natural language augmentations. In this work, we propose LLM-TTA, which uses LLM-generated augmentations as TTA's augmentation function. LLM-TTA outperforms conventional augmentation functions across sentiment, toxicity, and news classification tasks for BERT and T5 models, with BERT's OOD robustness improving by an average of 4.48 percentage points without regressing average ID performance. We explore selectively augmenting inputs based on prediction entropy to reduce the rate of expensive LLM augmentations, allowing us to maintain performance gains while reducing the average number of generated augmentations by 57.74\%. LLM-TTA is agnostic to the task model architecture, does not require OOD labels, and is effective across low and high-resource settings. We share our data, models, and code for reproducibility.

cs.LG↗

arXiv2024

Tensor network influence functionals in the continuous-time limit: connections to quantum embedding, bath discretization, and higher-order time propagation

We describe two developments of tensor network influence functionals (in particular, influence functional matrix product states (IF-MPS)) for quantum impurity dynamics within the fermionic setting of the Anderson impurity model. The first provides the correct extension of the IF-MPS to continuous time by introducing a related mathematical object, the boundary influence functional MPS. The second connects the dynamics described by a compressed IF-MPS to that of a quantum embedding method with a time-dependent effective bath undergoing nonunitary dynamics. Using these concepts, we implement higher-order time propagators for the quench dynamics of the Anderson impurity model within the boundary IF-MPS formalism. The calculations illustrate the ability of the current formulation to efficiently remove the time step error in standard discrete-time IF-MPS implementations as well as to interface with state vector propagation techniques. They also show the advantages of IF-MPS dynamics, with its associated highly compact effective bath dynamics, over state vector propagation with a static bath discretization.

cond-mat.str-el↗

arXiv2023

A discrete mean value of the Riemann zeta function

In this work, we estimate the sum \begin{align*} \sum_{0 < \Im(\rho) \leq T} \zeta(\rho+\alpha)X(\rho) Y(1\!-\! \rho) \end{align*} over the nontirival zeros $\rho$ of the Riemann zeta funtion where $\alpha$ is a complex number with $\alpha\ll 1/\log T$ and $X(\cdot)$ and $Y(\cdot)$ are some Dirichlet polynomials. Moreover, we estimate the discrete mean value above for higher derivatives where $\zeta(\rho+\alpha)$ is replaced by $\zeta^{(m)}(\rho)$ for all $m\in\mathbb{N}$. The formulae we obtain generalize a number of previous results in the literature. As an application, assuming the Riemann Hypothesis we obtain the lower bound \begin{align*} \sum_{0 < \Im(\rho) < T} | \zeta^{(m)}(\rho)|^{2k} \gg T(\log T)^{k^2+2km+1} \quad \quad (k,m\in\mathbb{N}) \end{align*} which was previously known under the Generalized Riemann Hypothesis, in the case $m=1$.

math.NT↗

arXiv2023

Mean values of long Dirichlet polynomials with divisor coefficients

In this article, we prove an asymptotic formula for the mean value of long smoothed Dirichlet polynomials with divisor coefficients. Our result has a main term that includes all lower order terms and a power saving error term. This is derived from a more general theorem on mean values of long smoothed Dirichlet polynomials that was previously established by the second and third authors. We thus establish a stronger form of a conjecture of Conrey and Gonek in the case of divisor functions.

math.NT↗

arXiv2023

Blind Biological Sequence Denoising with Self-Supervised Set Learning

Biological sequence analysis relies on the ability to denoise the imprecise output of sequencing platforms. We consider a common setting where a short sequence is read out repeatedly using a high-throughput long-read platform to generate multiple subreads, or noisy observations of the same sequence. Denoising these subreads with alignment-based approaches often fails when too few subreads are available or error rates are too high. In this paper, we propose a novel method for blindly denoising sets of sequences without directly observing clean source sequence labels. Our method, Self-Supervised Set Learning (SSSL), gathers subreads together in an embedding space and estimates a single set embedding as the midpoint of the subreads in both the latent and sequence spaces. This set embedding represents the "average" of the subreads and can be decoded into a prediction of the clean sequence. In experiments on simulated long-read DNA data, SSSL methods denoise small reads of $\leq 6$ subreads with 17% fewer errors and large reads of $>6$ subreads with 8% fewer errors compared to the best baseline. On a real dataset of antibody sequences, SSSL improves over baselines on two self-supervised metrics, with a significant improvement on difficult small reads that comprise over 60% of the test set. By accurately denoising these reads, SSSL promises to better realize the potential of high-throughput DNA sequencing data for downstream scientific applications.

q-bio.GN↗

arXiv2022

Tuning the Tail Latency of Distributed Queries Using Replication

Querying graph data with low latency is an important requirement in application domains such as social networks and knowledge graphs. Graph queries perform multiple hops between vertices. When data is partitioned and stored across multiple servers, queries executing at one server often need to hop to vertices stored by another server. Such distributed traversals represent a performance bottleneck for low-latency queries. To reduce query latency, one can replicate remote data to make distributed traversals unnecessary, but replication is expensive and should be minimized. In this paper, we introduce the problem of finding data replication schemes that satisfy arbitrary user-defined query latency constraints with minimal replication cost. We propose a novel workload model to express data access causality, propose a family of heuristics, and introduce non-trivial sufficient conditions for their correctness. Our evaluation on two representative benchmarks show that our algorithms enable fine-tuning query latency with data replication and can find sweet spots in the latency/replication design space.

cs.DB↗

arXiv2022

Real time evolution of Anderson impurity models via tensor network influence functionals

In this work we present and analyze two tensor network-based influence functional approaches for simulating the real-time dynamics of quantum impurity models such as the Anderson model. Via comparison with recent numerically exact simulations, we show that such methods accurately capture the long-time non-equilibrium quench dynamics. The two parameters that must be controlled in these tensor network influence functional approaches are a time discretization (Trotter) error and a bond dimension (tensor network truncation) error. We show that the actual numerical uncertainties are controlled by an intricate interplay of these two approximations which we demonstrate in different regimes. Our work opens the door to using these tensor network influence functional methods as general impurity solvers.

cond-mat.str-el↗

arXiv2022

If Influence Functions are the Answer, Then What is the Question?

Influence functions efficiently estimate the effect of removing a single training data point on a model's learned parameters. While influence estimates align well with leave-one-out retraining for linear models, recent works have shown this alignment is often poor in neural networks. In this work, we investigate the specific factors that cause this discrepancy by decomposing it into five separate terms. We study the contributions of each term on a variety of architectures and datasets and how they vary with factors such as network width and training time. While practical influence function estimates may be a poor match to leave-one-out retraining for nonlinear networks, we show they are often a good approximation to a different object we term the proximal Bregman response function (PBRF). Since the PBRF can still be used to answer many of the questions motivating influence functions, such as identifying influential or mislabeled examples, our results suggest that current algorithms for influence function estimation give more informative results than previous error analyses would suggest.

cs.LG↗