Zero Density Research Articles

An unconditional GLn large sieve

Let Fn be the set of cuspidal automorphic representations π of GLn over a number field with unitary central character. We prove two unconditional large sieve inequalities for the Hecke eigenvalues of π∈Fn, one on the integers and one on the primes. The second leads to the first unconditional zero density estimate for the family of L-functions L(s,π) associated to π∈Fn, which we make log-free. As an application of the zero density estimate, we prove a hybrid subconvexity bound for L(12,π) for a density one subset of π∈Fn.

Type-I contributions to the one and two level densities of quadratic Dirichlet L–functions over function fields

Using the Ratios Conjecture, we write down precise formulas with lower order terms for the one and the two level densities of zeros of quadratic Dirichlet L–functions over function fields. We denote the various terms arising as Type-0, Type-I and Type-II contributions. When the support of the Fourier transform of the test function is sufficiently restricted, we rigorously compute the Type-0 and Type-I terms and confirm that they match the conjectured answer. When the restrictions on the support are relaxed, our results suggest that Type-II contributions become important in the two level density.

Recommended Values for the Viscosity in the Limit of Zero Density and its Initial Density Dependence for Twelve Gases and Vapors: Revisited from Experiment between 297 K and 691 K

Previously published experimental viscosity data at low density, originally obtained using all-quartz oscillating-disk viscometers for 12 gases and vapors in the temperature range between 297 K and 691 K, were re-evaluated after an improved re-calibration. The relative combined expanded (k = 2) uncertainty of the re-evaluated data is 0.2% near room temperature and increases to 0.3% at higher temperatures. The re-evaluated data for sulfur hexafluoride, methanol, n-pentane, n-hexane, n-heptane, neopentane, cyclohexane, benzene, toluene, p-xylene, phenol, and triethylamine were arranged in approximately isothermal groups and converted into quasi-isothermal viscosity data using a first-order Taylor series in temperature. Then, they were evaluated by means of a series expansion truncated at first order to obtain the zero-density and initial density viscosity coefficients, η(0) and η(1). When the number of isothermal data or their quality was not adequate, the Rainwater–Friend theory for the initial density dependence of the viscosity was additionally used to derive η(0) and η(1) values. Finally, reliable η(0) and η(1) values, preferably obtained from the isotherms, were recommended as reference values for the 12 gases and vapors in the measured temperature range to be applied when generating any new viscosity formulation.

Characteristics of plasma parameters and turbulence in the isotope-mixing and the non-mixing states in hydrogen–deuterium mixture plasmas in the large helical device

Characteristics of plasma parameters and turbulence in the isotope-mixing and the non-mixing states in hydrogen-deuterium mixture plasmas in the large helical device are discussed. The isotope mixing state is characterized by the uniform isotope ratio profile regardless of the location of the particle source of each species in the isotope mixture plasma. The isotope non-mixing state is identified by the non-uniform isotope ratio profile measured with bulk charge exchange spectroscopy when the beam fueling isotope species differs from the recycling isotope species. The effect of collisionality, ratio, sign of density gradient on transition between isotope mixing and non-mixing is discussed. The plasma parameters preferable for the non-mixing state are found to be lower collisionality, higher , and negative or zero density gradient (peaked or flat density profile). The time scale of transition from non-mixing to mixing is evaluated by the hydrogen and deuterium pellet injection near the plasma edge and is found to be less than 5 ms, which is much shorter than the particle confinement time. The strong correlation between isotope mixing and turbulence characteristics is observed. This strong correlation suggests the change in turbulence is a strong candidate for the mechanism causing the transition between uniform and non-uniform isotope density ratio profiles.

Dynamical irreducibility of polynomials modulo primes

For a class of polynomials $$f \in {\mathbb {Z}}[X]$$ , which in particular includes all quadratic polynomials, and also trinomials of some special form, we show that, under some natural conditions (necessary for quadratic polynomials), the set of primes p such that all iterations of f are irreducible modulo p is of relative density zero. Furthermore, we give an explicit bound on the rate of the decay of the density of such primes in an interval [1, Q] as $$Q \rightarrow \infty $$ . For this class of polynomials this gives a more precise version of a recent result of Ferraguti (Proc Am Math Soc 146:2773–2784, 2018), which applies to arbitrary polynomials but requires a certain assumption about their Galois group. Furthermore, under the Generalised Riemann Hypothesis we obtain a stronger bound on this density.

INTRINSIC SMALLNESS

AbstractRecent work in computability theory has focused on various notions of asymptotic computability, which capture the idea of a set being “almost computable.” One potentially upsetting result is that all four notions of asymptotic computability admit “almost computable” sets in every Turing degree via coding tricks, contradicting the notion that “almost computable” sets should be computationally close to the computable sets. In response, Astor introduced the notion of intrinsic density: a set has defined intrinsic density if its image under any computable permutation has the same asymptotic density. Furthermore, introduced various notions of intrinsic computation in which the standard coding tricks cannot be used to embed intrinsically computable sets in every Turing degree. Our goal is to study the sets which are intrinsically small, i.e. those that have intrinsic density zero. We begin by studying which computable functions preserve intrinsic smallness. We also show that intrinsic smallness and hyperimmunity are computationally independent notions of smallness, i.e. any hyperimmune degree contains a Turing-equivalent hyperimmune set which is “as large as possible” and therefore not intrinsically small. Our discussion concludes by relativizing the notion of intrinsic smallness and discussing intrinsic computability as it relates to our study of intrinsic smallness.

The QCD phase diagram and heavy ion collisions from lattice QCD

The QCD phase diagram is an important ingrediant to understand both the develpment of the early universe and the results of recent heavy ion collison experiments. At zero baryon density lattice QCD is an established tool, that provides precise theoretical results. Calculations at non zero densities however require new techniques to deal with the sign problem. In this talk I will review our recent effort to investigate QCD at non-vanishing baryon chemical potential.

The density of complex zeros of random sums

Let {ηj}j=0N be a sequence of independent and identically distributed random complex Gaussian variables, and let {fj(z)}j=0N be a sequence of given analytic functions that are real-valued on the real line. We prove an exact formula for the expected density of the distribution of complex zeros of the random equation ∑j=0Nηjfj(z)=K, where K∈C. The method of proof employs a formula for the expected absolute value of quadratic forms of Gaussian random variables. We also obtain the limiting behaviour of the density function as N tends to infinity and provide numerical computations for the density function and empirical distributions for random sums with various choices of the functions fj(z). Finally, we study the cases when the functions fj(z) are polynomials orthogonal on the real line and the unit circle.

Divisor sums representable as the sum of two squares

Let $s(n)$ denote the sum of the proper divisors of the natural number $n$. We show that the number of $n \leq x$ such that $s(n)$ is a sum of two squares has order of magnitude $x/\sqrt{\log x}$, which agrees with the count of $n \leq x$ which are a sum of two squares. Our result confirms a special case of a conjecture of Erd{\H o}s, Granville, Pomerance and Spiro, who in a 1990 paper asserted that if $\mathcal{A} \subset \mathbb{N}$ has asymptotic density zero (e.g. if $\mathcal{A}$ is the set of $n \leq x$ which are a sum of two squares), then $s^{-1}(\mathcal{A})$ also has asymptotic density zero.

Explicit zero density estimate for the Riemann zeta-function near the critical line

In 1946, A. Selberg proved N(σ,T)≪T1−14(σ−12)log⁡T where N(σ,T) is the number of nontrivial zeros ρ of the Riemann zeta-function with ℜ{ρ}>σ and 0<ℑ{ρ}≤T. We provide an explicit version of this estimate, together with an explicit approximate functional equation and an explicit upper bound for the second power moment of the zeta-function on the critical line.

Partial inverse nodal problems for differential pencils on a star‐shaped graph

In this paper, the authors study partial inverse nodal problems for differential pencils on a star‐shaped graph. We firstly show that the potential on each edge can be uniquely determined by twin‐dense nodal subsets on some interior intervals under certain conditions. Without any nodal information on some potential on the fixed edge, we may add some spectral information to guarantee these uniqueness theorems. We still consider the case of arbitrary intervals having the internal vertex. In particular, we pose and solve a new partial inverse nodal problem for differential pencils on the star‐shaped graph from the Weyl m‐function and the theory concerning densities of zeros of entire functions.

N-level density of the low-lying zeros of primitive Dirichlet L-functions

Katz and Sarnak conjectured that the statistics of low-lying zeros of various family of L-functions matched with the scaling limit of eigenvalues from the random matrix theory. In this paper we confirm this statistic for a family of primitive Dirichlet L-functions matches up with corresponding statistic in the random unitary ensemble, in a range that includes the off-diagonal contribution. To estimate the n-level density of zeros of the L-functions, we use the asymptotic large sieve method developed by Conrey, Iwaniec and Soundararajan. For the random matrix side, a formula from Conrey and Snaith allows us to solve the matchup problem.

Distinguishing density and the Distinct Spheres Condition

If a graph G has distinguishing number 2, then there exists a partition of its vertex set into two parts, such that no nontrivial automorphism of G fixes setwise the two parts. Such a partition is called a 2-distinguishing coloring of G, and the parts are called its color classes. If G admits such a coloring, it is often possible to find another in which one of the color classes is sparse in a certain sense. In this case we say that G has 2-distinguishing density zero. An extreme example of this would be an infinite graph admitting a 2-distinguishing coloring in which one of the color classes is finite.The Infinite Motion Conjecture is a well-known open conjecture about 2-distinguishability. A graph G is said to have infinite motion if every nontrivial automorphism of G moves infinitely many vertices, and the conjecture states that every connected, locally finite graph with infinite motion is 2-distinguishable. In this paper we show that for many classes of graphs for which the Infinite Motion Conjecture is known to hold, the graphs have 2-distinguishing density zero.

On the size of primitive sets in function fields

A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular, we show that primitive sets in the function field have lower density zero by showing that the sum ∑a∈A1qdeg⁡adeg⁡a, an analogue of a sum considered by Erdős, is uniformly bounded over all primitive sets A. We then adapt a method of Besicovitch to construct primitive sets in Fq[x] with upper density arbitrarily close to q−1q and generalize a result of Martin and Pomerance on the asymptotic growth rate of the counting function of a primitive set. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for function fields, as well as bounds on the size of the k-th irreducible polynomial.

The density zero ideal and the splitting number

The main result of this paper is an improvement of the upper bound on the cardinal invariant cov⁎(Z0) that was discovered in [11]. Here Z0 is the ideal of subsets of the set of natural numbers that have asymptotic density zero. This improved upper bound is also dualized to get a better lower bound on the cardinal non⁎(Z0). En route some variations on the splitting number are introduced and several relationships between these variants are proved.

The logarithmic derivative of entire functions with $\upsilon$-density of zeros along curves of regular rotation

The logarithmic derivative of entire functions with $\upsilon$-density of zeros along curves of regular rotation

Zeros of partition function for continuous phase transitions using cumulants

This paper explores the use of a cumulant method to determine the zeros of partition functions for continuous phase transitions going up to the second zero. Unlike a first-order transition, with a uniform density of zeros near the transition point, a continuous transition shows a power law dependence of the density with a nontrivial slope for the line of zeros. Different types of models and methods of generating cumulants are used as testing grounds for the method. These include exactly solvable DNA melting problem on hierarchical lattices, heterogeneous DNA melting with randomness in sequence, Monte Carlo simulations for the well-known square lattice Ising model. The method is applicable for zeros near the imaginary axis which control dynamical quantum phase transitions. In all cases, the method is found to provide the basic information about the transition, and most importantly, avoids root finding methods.

On the interplay between additive and multiplicative largeness and its combinatorial applications

Many natural notions of additive and multiplicative largeness arise from results in Ramsey theory. In this paper, we explain the relationships between these notions for subsets of N and in more general ring-theoretic structures. We show that multiplicative largeness begets additive largeness in three ways and give a collection of examples demonstrating the optimality of these results. We also give a variety of applications arising from the connection between additive and multiplicative largeness. For example, we show that given any n,k∈N, any finite set with fewer than n elements in a sufficiently large finite field can be translated so that each of its elements becomes a non-zero kth power. We also prove a theorem concerning Diophantine approximation along multiplicatively syndetic subsets of N and a theorem showing that subsets of positive upper Banach density in certain multiplicative sub-semigroups of N of zero density contain arbitrarily long arithmetic progressions. Along the way, we develop a new characterization of upper Banach density in a wide class of amenable semigroups and make explicit the uniformity in recurrence theorems from measure theoretic and topological dynamics. This in turn leads to strengthened forms of classical theorems of Szemerédi and van der Waerden on arithmetic progressions.

Dense quark\u2013gluon plasma in strong magnetic fields

A non-perturbative (np) method of Field correlators (FCM) was applied to study QCD at temperatures above the deconfinement transition (1<T/T_c<3,~T_csim 0.16~mathrm{GeV}) and nonzero baryon densities (baryon chemical potential mu _B<0.5~mathrm{GeV}) in an external uniform magnetic field (eB<0.5~mathrm{GeV}^2). Within FCM, the np high-temperature dynamics is embodied in the Polyakov loop and in the Debye mass due to the Color-Magnetic confinement. Analytic expressions for quark pressure and magnetic susceptibility were obtained. The expressions were represented as series and in integral form. Magnetic susceptibility was found to increase rapidly with temperature and slowly with density. The results at the zero density limit are in agreement with lattice data.

Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density

We study the Cauchy problem of nonhomogeneous magneto-micropolar fluid system with zero density at infinity in the entire space [Formula: see text]. We prove that the system admits a unique local strong solution provided the initial density and the initial magnetic field decay not too slowly at infinity. In particular, there is no need to require any Choe–Kim type compatibility condition for the initial data.