A quick route to unique factorization in quadratic orders
We give a short proof -- not relying on ideal classes or the geometry of numbers -- of a known criterion for quadratic orders to possess unique factorization.
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arXiv subjects
Paul Pollack
Publications and source records attributed to Paul Pollack.
The reciprocal sum of divisors of Mersenne numbers
We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine $\max_{n\le x} \sum_{p \mid 2^n-1} 1/p$ to within $o(1)$ and $\max_{n\le x} \sum_{d\mid 2^n-1}1/d$ to within a factor of $1+o(1)$, as $x\to\infty$. This refines, conditionally, earlier estimates of Erdős and Erdős-Kiss-Pomerance. Conditionally (only) on GRH, we also determine $\sum 1/d$ to within a factor of $1+o(1)$ where $d$ runs over all numbers dividing $2^n-1$ for some $n\le x$. This conditionally confirms a conjecture of Pomerance and answers a question of Murty-Rosen-Silverman. Finally, we show that both $\sum_{p\mid 2^n-1} 1/p$ and $\sum_{d\mid 2^n-1}1/d$ admit continuous distribution functions in the sense of probabilistic number theory.
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Numbers which are orders only of cyclic groups
We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,ϕ(n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$, Erdős proved that $$C(x) \sim e^{-γ} x/\log\log\log{x}, \quad\text{as $x\to\infty$}.$$ We show that $C(x)$ has an asymptotic series expansion, in the sense of Poincaré, in descending powers of $\log\log\log{x}$, namely $$\frac{e^{-γ} x}{\log\log\log{x}} \left(1-\fracγ{\log\log\log{x}} + \frac{γ^2 + \frac{1}{12}π^2}{(\log\log\log{x})^2} - \frac{γ^3 +\frac{1}{4} γπ^2 + \frac{2}{3}ζ(3)}{(\log\log\log{x})^3} + \dots \right). $$
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Finite sets containing near-primitive roots
Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $ε> 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-ε}$. It is an open problem to show the same result with $\frac12$ replaced by any larger constant. We show that if $a,b$ are multiplicatively independent, then for almost all primes $p$, one of $a,b,ab, a^2b, ab^2$ has order exceeding $p^{\frac{1}{2}+\frac{1}{30}}$. The same method allows one to produce, for each $ε> 0$, explicit finite sets $\mathcal{A}$ with the property that for almost all primes $p$, some element of $\mathcal{A}$ has order exceeding $p^{1-ε}$. Similar results hold for orders modulo general integers $n$ rather than primes $p$.
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Phi, Primorials, and Poisson
The primorial $p\#$ of a prime $p$ is the product of all primes $q\le p$. Let pr$(n)$ denote the largest prime $p$ with $p\# \mid ϕ(n)$, where $ϕ$ is Euler's totient function. We show that the normal order of pr$(n)$ is $\log\log n/\log\log\log n$. That is, pr$(n) \sim \log\log n/\log\log\log n$ as $n\to\infty$ on a set of integers of asymptotic density 1. In fact we show there is an asymptotic secondary term and, on a tertiary level, there is an asymptotic Poisson distribution. We also show an analogous result for the largest integer $k$ with $k!\mid ϕ(n)$.
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Symmetric primes revisited
A pair of odd primes is said to be symmetric if each prime is congruent to one modulo their difference. A theorem from 1996 by Fletcher, Lindgren, and the third author provides an upper bound on the number of primes up to x that belong to a symmetric pair. In the present paper, that theorem is improved to what is likely to be the best possible result. We also establish that there exist infinitely many symmetric pairs of primes. In fact, we show that for every integer m at least 2 there is a string of m consecutive primes, any two of which form a symmetric pair.
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Twists of hyperelliptic curves by integers in progressions modulo $p$
Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.
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Counting and effective rigidity in algebra and geometry
The purpose of this article is to produce effective versions of some rigidity results in algebra and geometry. On the geometric side, we focus on the spectrum of primitive geodesic lengths (resp., complex lengths) for arithmetic hyperbolic 2-manifolds (resp., 3-manifolds). By work of Reid, this spectrum determines the commensurability class of the 2-manifold (resp., 3-manifold). We establish effective versions of these rigidity results by ensuring that, for two incommensurable arithmetic manifolds of bounded volume, the length sets (resp., the complex length sets) must disagree for a length that can be explicitly bounded as a function of volume. We also prove an effective version of a similar rigidity result established by the second author with Reid on a surface analog of the length spectrum for hyperbolic 3-manifolds. These effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras. To prove these effective rigidity results, we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest.
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Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach
Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q<2p^{1/2}$ that is a cubic residue modulo $p$. Here we show that for each fixed $ε> 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(ε)$, the number of prime cubic residues $q < p^{1/2+ε}$ exceeds $p^{ε/30}$. Our argument, like Nagell's, is rooted in the law of cubic reciprocity; somewhat surprisingly, character sum estimates play no role. We use the same method to establish related results about prime quadratic and biquadratic residues. For example, for all large primes $p$, there are more than $p^{1/9}$ prime quadratic residues $q<p$.
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Typically bounding torsion
We formulate the notion of \emph{typical boundedness} of torsion on a family of abelian varieties defined over number fields. This means that the torsion subgroups of elements in the family can be made uniformly bounded by removing from the family all abelian varieties defined over number fields of degree lying in a set of arbitrarily small density. We show that for each fixed $g$, torsion is typically bounded on the family of all $g$-dimensional CM abelian varieties. We show that torsion is \emph{not} typically bounded on the family of all elliptic curves, and we establish results -- some unconditional and some conditional -- on typical boundedness of torsion of elliptic curves for which the degree of the $j$-invariant is fixed.
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Divisor-sum fibers
Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erdős-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $ε(x)$ is any function tending to $0$ as $x\to\infty$, and $\mathcal{A}$ is a set of integers of cardinality at most $x^{\frac12+ε(x)}$, then the number of integers $n\le x$ with $s(n) \in \mathcal{A}$ is $o(x)$, as $x\to\infty$. In particular, the EGPS conjecture holds for infinite sets with counting function $O(x^{\frac12 + ε(x)})$. We also disprove a hypothesis from the same paper of EGPS by showing that for any positive numbers $α$ and $ε$, there are integers $n$ with arbitrarily many $s$-preimages lying between $α(1-ε)n$ and $α(1+ε)n$. Finally, we make some remarks on solutions $n$ to congruences of the form $σ(n) \equiv a\pmod{n}$, proposing a modification of a conjecture appearing in recent work of the first two authors. We also improve a previous upper bound for the number of solutions $n \leq x$, making it uniform in $a$.
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Pursuing polynomial bounds on torsion
We show that for all epsilon > 0, there is a constant C(epsilon) > 0 such that for all elliptic curves E defined over a number field F with j(E) in Q we have #E(F)[tors] \leq C(epsilon)[F:Q]^{5/2+epsilon}. We pursue further bounds on the size of the torsion subgroup of an elliptic curve over a number field E/F that are polynomial in [F:Q] under restrictions on j(E). We give an unconditional result for j(E) lying in a fixed quadratic field that is not imaginary of class number one as well as two further results, one conditional on GRH and one conditional on the strong boundedness of isogenies of prime degree for non-CM elliptic curves.
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Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds
In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first $m$ geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset $S$ of its geodesic length spectrum, and any $k \geq 2$, we produce infinitely many $k$-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing $S$, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.
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Refinements of Lagrange's four-square theorem
A well-known theorem of Lagrange asserts that every nonnegative integer $n$ can be written in the form $a^2+b^2+c^2+d^2$, where $a,b,c,d \in \mathbb{Z}$. We characterize the values assumed by $a+b+c+d$ as we range over all such representations of $n$.
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Systoles of Arithmetic Hyperbolic Surfaces and 3-manifolds
Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the set of all commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with invariant trace field $k$. The proof relies upon bounds for the absolute logarithmic Weil height of algebraic integers due to Silverman, Brindza and Hajdu, as well as precise estimates for the number of rational quaternion algebras not admitting embeddings of any quadratic field having small discriminant. When the trace field is $\mathbf{Q}$, using work of Granville and Soundararajan, we establish a stronger result that allows our constant lower bound $x_0$ to grow with the area. As an application, we establish a systolic bound for arithmetic hyperbolic surfaces that is related to prior work of Buser-Sarnak and Katz-Schaps-Vishne. Finally, we establish an analogous density result for commensurability classes of arithmetic hyperbolic 3-orbifolds with small area totally geodesic $2$-orbifolds.
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The truth about torsion in the CM case, II
Let $T_{\rm CM}(d)$ be the largest size of the torsion subgroup of an elliptic curve with complex multiplication (CM) defined over a degree $d$ number field. Work of Breuer and Clark--Pollack showed $\limsup_{d \to \infty} \frac{T_{\rm CM}(d)}{d \log \log d} \in (0,\infty)$. Here we show that the above limit supremum is precisely $\frac{e^γ π}{\sqrt{3}}$. We also study -- in part, out of necessity -- the upper order of the size of the torsion subgroup of various restricted classes of CM elliptic curves over number fields.
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Two problems concerning irreducible elements in rings of integers of number fields
Let $K$ be a number field with ring of integers $\mathbb{Z}_K$. We prove two asymptotic formulas connected with the distribution of irreducible elements in $\mathbb{Z}_K$. First, we estimate the maximum number of nonassociated irreducibles dividing a nonzero element of $\mathbb{Z}_K$ of norm not exceeding $x$ (in absolute value), as $x\to\infty$. Second, we count the number of irreducible elements of $\mathbb{Z}_K$ of norm not exceeding $x$ lying in a given arithmetic progression (again, as $x\to\infty$). When $K=\mathbb{Q}$, both results are classical; a new feature in the general case is the influence of combinatorial properties of the class group of $K$.
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An elemental Erdős-Kac theorem for algebraic number fields
Fix a number field $K$. For each nonzero $α\in \mathbb{Z}_K$, let $ν(α)$ denote the number of distinct, nonassociate irreducible divisors of $α$. We show that $ν(α)$ is normally distributed with mean proportional to $(\log\log |N(α)|)^{D}$ and standard deviation proportional to $(\log\log{|N(α)|})^{D-1/2}$. Here $D$, as well as the constants of proportionality, depend only on the class group of $K$. For example, for each fixed real $λ$, the proportion of $α\in \mathbb{Z}[\sqrt{-5}]$ with $$ ν(α) \le \frac{1}{8}(\log\log{N(α)})^2 + \fracλ{2\sqrt{2}} (\log\log{N(α)})^{3/2} $$ is given by $\frac{1}{\sqrt{2π}} \int_{-\infty}^λ e^{-t^2/2}\, \mathrm{d}t$. As further evidence that "irreducibles play a game of chance", we show that the values $ν(α)$ are equidistributed modulo $m$ for every fixed $m$.
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