arXiv subjects
Publications and source records attributed to Nathan Ng.
In this article, we prove a new bound for the least prime ideal in the Chebotarev density theorem, which improves the main theorem of Zaman [Funct. Approx. Comment. Math. 57 (2017), no.1, 115-142] by a factor of $5/2$. Our main improvement comes from a new version of Turán's power sum method. The key new idea is to use Harnack's inequality for harmonic functions to derive a superior lower bound for the generalised Fejér kernel.
We generalize a method of Conrey and Ghosh (Invent. Math. 94 (1988)) to prove quantitative estimates for simple zeros of modular form L-functions of arbitrary conductor.
Modifying a method of Jutila, we prove a t aspect subconvexity estimate for L-functions associated to primitive holomorphic cusp forms of arbitrary level that is of comparable strength to Good's bound for the full modular group, thus resolving a problem that has been open for 35 years. A key innovation in our proof is a general form of Voronoi summation that applies to all fractions, even when the level is not squarefree.
Scheduling surgeries is a challenging task due to the fundamental uncertainty of the clinical environment, as well as the risks and costs associated with under- and over-booking. We investigate neural regression algorithms to estimate the parameters of surgery case durations, focusing on the issue of heteroscedasticity. We seek to simultaneously estimate the duration of each surgery, as well as a surgery-specific notion of our uncertainty about its duration. Estimating this uncertainty can lead to more nuanced and effective scheduling strategies, as we are able to schedule surgeries more efficiently while allowing an informed and case-specific margin of error. Using surgery records %from the UC San Diego Health System, from a large United States health system we demonstrate potential improvements on the order of 20% (in terms of minutes overbooked) compared to current scheduling techniques. Moreover, we demonstrate that surgery durations are indeed heteroscedastic. We show that models that estimate case-specific uncertainty better fit the data (log likelihood). Additionally, we show that the heteroscedastic predictions can more optimally trade off between over and under-booking minutes, especially when idle minutes and scheduling collisions confer disparate costs.
Additive divisor sums play a prominent role in the theory of the moments of the Riemann zeta function. There is a long history of determining sharp asymptotic formula for the shifted convolution sum of the ordinary divisor function. In recent years, it has emerged that a sharp asymptotic formula for the shifted convolution sum of the triple divisor function would be useful in evaluating the sixth moment of the Riemann zeta function. In this article, we study $D_{k,\ell}(x) = \sum_{n \le x} τ_k(n) τ_{\ell}(n+h)$ where $τ_k$ and $τ_{\ell}$ are the $k$-th and $\ell$-th divisor functions. The main result is a lower bound of the correct order of magnitude for $D_{k,\ell}(x,h)$, uniform in $h$. In addition, the conjectural asymptotic formula for $D_{k,\ell}(x,h)$ is studied. Using an argument of Ivić and Conrey-Gonek the leading term in the conjectural asymptotic formula is simplified. In addition, a probabilistic method is presented which gives the same leading term. Finally, we show that these two methods give the same answer as in a recent probabilistic argument of Terry Tao.
We present a numerical solution of the quantum Lenard-Balescu equation using a spectral method, namely an expansion in Laguerre polynomials. This method exactly conserves both particles and energy and facilitates the integration over the dielectric function. To demonstrate the method, we solve the equilibration problem for a spatially homogeneous one-component plasma with various initial conditions. Unlike the more usual Landau/Fokker-Planck system, this method requires no input Coulomb logarithm; the logarithmic terms in the collision integral arise naturally from the equation along with the non-logarithmic order-unity terms. The spectral method can also be used to solve the Landau equation and a quantum version of the Landau equation in which the integration over the wavenumber requires only a lower cutoff. We solve these problems as well and compare them with the full Lenard-Balescu solution in the weak-coupling limit. Finally, we discuss the possible generalization of this method to include spatial inhomogeneity and velocity anisotropy.
We show that ions at the low densities and high magnetic fields relevant to the outer crust of magnetars form a novel crystalline phase where ions are strongly coupled along the magnetic field and loosely coupled in the transverse direction. The underlying cause is the anisotropic screening of the Coulomb force by electrons in the presence of a strongly quantizing magnetic field which leads to Friedel oscillations in the ion-ion potential. In particular, the Friedel oscillations are much longer-ranged in the direction of the magnetic field than is the case in the absence of magnetic fields, a factor that has been neglected in previous studies. These "Friedel crystals" have very anisotropic elastic moduli, with potentially interesting implications for the Quasi-periodic Oscillations seen in the X-ray flux of magnetars during their giant flares. We find the minimum energy configuration of ions taking into account these anisotropic effects and find that, depending on the density, temperature and magnetic field strength, different lattice structures (fcc, hcp or bcc oriented in different ways in relation to the magnetic field) are favored.
In this article we prove a general theorem which establishes the existence of limiting distributions for a wide class of error terms from prime number theory. As a corollary to our main theorem, we deduce previous results of Wintner (1935), Rubinstein and Sarnak (1994), and of Ng (2004). In addition, we establish limiting distribution results for the error term in the prime number theorem for an automorphic $L$-function, weighted sums of the Möbius function, weighted sums of the Liouville function, the sum of the Möbius function in an arithmetic progression, and the error term in Chebotarev's density theorem.
Assuming the generalized Riemann hypothesis, we prove quantitative estimates for the number of simple zeros on the critical line for the L-functions attached to classical holomorphic newforms.
Let $L(s,χ)$ be a fixed Dirichlet $L$-function. Given a vertical arithmetic progression of $T$ points on the line $\Re(s)=1/2$, we show that $\gg T \log T$ of them are not zeros of $L(s,χ)$. This result provides some theoretical evidence towards the conjecture that all ordinates of zeros of Dirichlet $L$-functions are linearly independent over the rationals. We also establish an upper bound (depending upon the progression) for the first member of the arithmetic progression that is not a zero of $L(s,χ)$.
This article studies the zeros of Dedekind zeta functions. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of the Deuring-Heilbronn phenomenon. In addition, we obtain an explicit bound for the number of zeros in a box.
We consider the the effect of the electron cloud about a vorton in the CFL-$K^0$ high-density phase by numerically solving the ultrarelativistic Thomas-Fermi equation about a toroidal charge. Including electrons removes the electric monopole contribution to the energy, and noticeably decreases the equilibrium radius of these stable vortex loops.
We derive a lower bound for a second moment of the reciprocal of the derivative of the Riemann zeta-function averaged over the zeros of the zeta-function that is half the size of the conjectured value. Our result is conditional upon the assumption of the Riemann Hypothesis and the conjecture that the zeros of the zeta-function are simple.
Assuming the Riemann Hypothesis, we establish lower bounds for moments of the derivative of the Riemann zeta-function averaged over the non-trivial zeros of $ζ(s)$. Our proof is based upon a recent method of Rudnick and Soundararajan that provides analogous bounds for moments of $L$-functions at the central point, averaged over families.
In this article we compute a discrete mean value of the derivative of the Riemann zeta function. This mean value will be important for several applications concerning the size of $ζ'(ρ)$ where $ζ(s)$ is the Riemann zeta function and $ρ$ is a non-trivial zero of the Riemann zeta function.
In this article we exhibit small and large values of $ζ'(ρ)$ by applying Soundararajan's resonance method. Our results assume the Riemann hypothesis.
We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the Riemann zeta function whose spacing is 2.9125 times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.
Let the summatory function of the Möbius function be denoted $M(x)$. We deduce in this article conditional results concerning $M(x)$ assuming the Riemann Hypothesis and a conjecture of Gonek and Hejhal on the negative moments of the Riemann zeta function. The main results shown are that the weak Mertens conjecture and the existence of a limiting distribution of $e^{-y/2}M(e^{y})$ are consequences of the aforementioned conjectures. By probabilistic techniques, we present an argument that suggests $M(x)$ grows as large positive and large negative as a constant times $\pm \sqrt{x} (\log \log \log x)^{5/4}$ infinitely often, thus providing evidence for an unpublished conjecture of Gonek's.