arXiv subjects

Jeffrey C. Lagarias

Publications and source records attributed to Jeffrey C. Lagarias.

At least 37 records · Page 2Linked to original sources

arXiv2018

A family of measures on symmetric groups and the field with one element

For each positive integer n this paper considers a one-parameter family of complex-valued measures on the symmetric group S_n, depending on a complex parameter z. For parameter values z=p^f a prime power, this measure describes splitting probabilities for monic degree n polynomials over the finite field with p^f elements, conditioned on being square-free. It studies these measure in the special case z=1, and shows they have an interesting internal structure having a representation theoretic interpretation. These measures may provide data relevant to the hypothetical "field with one element ${\mathbb F_1}$". It additionally studies the case z=-1, which also has a representation-theoretic interpretation

math.NT↗

arXiv2017

Shifted polyharmonic Maass forms for PSL(2,Z)

We study the vector space V_k^m(λ) of shifted polyharmonic Maass forms of weight k \in 2Z, depth m \geq 0, and shift λ\in C. This space is composed of real-analytic modular forms of weight k for PSL(2,Z) with moderate growth at the cusp which are annihilated by (Δ_k - λ)^m, where Δ_k is the weight k hyperbolic Laplacian. We treat the case λ\neq 0, complementing work of the second and third authors on polyharmonic Maass forms (with no shift). We show that V_k^m(λ) is finite-dimensional and bound its dimension. We explain the role of the real-analytic Eisenstein series E_k(z,s) with λ=s(s+k-1) and of the differential operator d/ds in this theory.

math.NT↗

arXiv2017

Polynomial splitting measures and cohomology of the pure braid group

We study for each $n$ a one-parameter family of complex-valued measures on the symmetric group $S_n$, which interpolate the probability of a monic, degree $n$, square-free polynomial in $\mathbb{F}_q[x]$ having a given factorization type. For a fixed factorization type, indexed by a partition $λ$ of $n$, the measure is known to be a Laurent polynomial. We express the coefficients of this polynomial in terms of characters associated to $S_n$-subrepresentations of the cohomology of the pure braid group $H^{\bullet}(P_n, \mathbb{Q})$. We deduce that the splitting measures for all parameter values $z= -\frac{1}{m}$ (resp. $z= \frac{1}{m}$), after rescaling, are characters of $S_n$-representations (resp. virtual $S_n$-representations.)

math.RT↗

arXiv2016

Dilated Floor Functions That Commute

We determine all pairs of real numbers $(α, β)$ such that the dilated floor functions $\lfloor αx\rfloor$ and $\lfloor βx\rfloor$ commute under composition, i.e., such that $\lfloor α\lfloor βx\rfloor\rfloor = \lfloor β\lfloor αx\rfloor\rfloor$ holds for all real $x$.

math.NT↗

arXiv2016

The Lerch zeta function IV. Hecke operators

This paper studies algebraic and analytic structures associated with the Lerch zeta function. It defines a family of two-variable Hecke operators $\{ T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1} f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions, including Lerch zeta functions for various parameter values. It determines the action of various related operators on these function spaces. It characterizes Lerch zeta functions (for fixed $s$ in the following way. It shows that there is for each $s \in {\bf C}$ a two-dimensional vector space spanned by linear combinations of Lerch zeta functions is characterized as a maximal space of simultaneous eigenfunctions for this family of Hecke operators. This result is an analogue of a result of Milnor for the Hurwitz zeta function. We also relate these functions to a linear partial differential operator in the $(a, c)$-variables having the Lerch zeta function as an eigenfunction.

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arXiv2015

Intersections of multiplicative translates of $3$-adic Cantor sets II: two infinite families

This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}Σ_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}Σ_{3,\bar{2}}$ for integers $1\leq M_1<\cdots<M_n$, where $Σ_{3,\bar{2}}$ denotes the $3$-adic Cantor set. This study was motivated by questions concerning the discrete dynamical system on the $3$-adic integers $\mathbb{Z}_3$ given by multiplication by $2$. The exceptional set $\mathcal{E}(\mathbb{Z}_3)$ is defined to be the set of all elements of $\mathbb{Z}_3$ whose forward orbits under this action intersect the $3$-adic Cantor set $Σ_{3,\bar{2}}$ infinitely many times. It is conjectured that it has Hausdorff dimension $0$. Part I showed that upper bounds on the Hausdorff dimension of the exceptional set can be extracted from knowing Hausdorff dimensions of sets of the kind above, in cases where all $M_i$ are powers of $2$. These intersection sets were shown to be fractals whose points have $3$-adic expansions describable by labeled paths in a finite automaton, whose Hausdorff dimension is exactly computable and is of the form $\log_3(β)$ where $β$ is a real algebraic integer. It gave algorithms for determination of the automaton, and computed examples showing that the dependence of the automaton and the value $β$ on the parameters $(M_1,\ldots,M_n)$ is complicated. The present paper studies two new infinite families of examples, illustrating interesting behavior of the automata and of the Hausdorff dimension of the associated fractals. One family has associated automata whose directed graph has a nested sequence of strongly connected components of arbitrarily large depth. The second family leads to an improved upper bound for the Hausdorff dimension of $\mathcal{E}(\mathbb{Z}_3)$ of $\log_3 ϕ\approx 0.438018$, where $ϕ$ denotes the Golden ratio.

math.NT↗

arXiv2015

Notes on Cardinal's Matrices

These notes are motivated by the work of Jean-Paul Cardinal on symmetric matrices related to the Mertens function. He showed that certain norm bounds on his matrices implied the Riemann hypothesis. Using a different matrix norm we show an equivalence of the Riemann hypothesis to suitable norm bounds on his matrices in the new norm. Then we specify a deformed version of his Mertens function matrices that unconditionally satisfies a norm bound that is of the same strength as his Riemann hypothesis bound.

math.NT↗

arXiv2015

The Lerch zeta function III. Polylogarithms and special values

This paper studies algebraic and analytic structures associated with the Lerch zeta function, extending the complex variables viewpoint taken in part II. The Lerch transcendent $Φ(s, z, c)$ is obtained from the Lerch zeta function $ζ(s, a, c)$ by the change of variable $z=e^{2 πi a}$. We show that it analytically continues to a maximal domain of holomorphy in three complex variables $(s, z, c)$ as a multivalued function defined over the base manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times ({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions defining the multivaluedness. For positive integer values s=m and c=1 this function is closely related to the classical m-th order polylogarithm $Li_m(z)$ We study its behavior as a function of two variables $(z, c)$ for special values where s=m is an integer. For $m \ge 1$ it gives a one-parameter deformation of the polylogarithm, and satisfies a linear ODE with coefficients depending on c, of order m+1 of Fuchsian type. We determine its (m+1)-dimensional monodromy representation, which is a non-algebraic deformation of the monodromy of $Li_m(z).$

math.NT↗

arXiv2015

Products of binomial coefficients and unreduced Farey fractions

This paper studies the product $\bar{G}_n$ of the binomial coefficients in the n-th row of Pascal's triangle, which equals the reciprocal of the product of all the reduced and unreduced Farey fractions of order n. It studies its size as a real number, measured by its logarithm $log(\bar{G}_n)$, and its prime factorization, measured by the order of divisibility by a fixed prime p, each viewed as a function of n. It derives three formulas for its prime power divisibility, $ord_p(\bar{G}_n)$, two of which relate it to base p radix expansions of n, and which display different facets of its behavior. These formulas are used to determine the maximal growth rate of each $ord_p(\bar{G}_n)$ and structure of the fluctuations of these functions. It also defines analogous functions for all integer bases $b$ replacing prime bases. A final topic relates the factorizations of $\bar{G}_n$ to Chebyshev-type prime-counting estimates and the prime number theorem.

math.NT↗

arXiv2015

Polyharmonic Maass forms for PSL(2, Z)

We discuss polyharmonic Maass forms of even integer weight on PSL(2, Z)\H, which are a generalization of classical Maass forms. We explain the role of real-analytic Eisenstein series E_k(z, s) and the differential operator $\partial/\partial s$ in this theory.

math.AP↗

arXiv2015

Splitting Behavior of $S_n$-Polynomials

We analyze the probability that, for a fixed finite set of primes S, a random, monic, degree n polynomial f(x) with integer coefficients in a box of side B around 0 satisfies: (i) f(x) is irreducible over the rationals, with splitting field over the rationals having Galois group $S_n$; (ii) the polynomial discriminant Disc(f) is relatively prime to all primes in S; (iii) f(x) has a prescribed splitting type at each prime p in S. The limit probabilities as $B \to \infty$ are described in terms of values of a one-parameter family of measures on $S_n$, called splitting measures, with parameter $z$ evaluated at the primes p in S. We study properties of these measures. We deduce that there exist degree n extensions of the rationals with Galois closure having Galois group $S_n$ with a given finite set of primes S having given Artin symbols, with some restrictions on allowed Artin symbols for p<n. We compare the distributions of these measures with distributions formulated by Bhargava for splitting probabilities for a fixed prime $p$ in such degree $n$ extensions ordered by size of discriminant, conditioned to be relatively prime to $p$.

math.NT↗

arXiv2014

$3x+1$ inverse orbit generating functions almost always have natural boundaries

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that $m$ is in the forward orbit of $n$ under iteration of $T_k(\cdot).$ We consider the generating functions $f_{k,m}(z) = \sum_{n>0, n \rightarrow m} z^{n},$ which are holomorphic in the unit disk. We give sufficient conditions on $(k,m)$ for the functions $f_{k, m}(z)$ have the unit circle $\{|z|=1\}$ as a natural boundary to analytic continuation. For the $3x+1$ function these conditions hold for all $m \ge 1$ to show that $f_{1,m}(z)$ has the unit circle as a natural boundary except possibly for $m= 1, 2, 4$ and $8$. The $3x+1$ Conjecture is equivalent to the assertion that $f_{1, m}(z)$ is a rational function of $z$ for the remaining values $m=1,2, 4, 8$.

math.CV↗

arXiv2013

A Skolem-Mahler-Lech theorem for iterated automorphisms of $K$-algebras

This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $σ$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $σ^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $σ\in {\rm Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $σ^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic.

math.AG↗

arXiv2013

Path sets in one-sided symbolic dynamics

Path sets are spaces of one-sided infinite symbol sequences associated to pointed graphs (G_v_0), which are edge-labeled directed graphs G with a distinguished vertex v_0. Such sets arise naturally as address labels in geometric fractal constructions and in other contexts. The resulting set of symbol sequences need not be closed under the one-sided shift. this paper establishes basic properties of the structure and symbolic dynamics of path sets, and shows they are a strict generalization of one-sided sofic shifts.

math.DS↗

arXiv2013

p-adic path set fractals and arithmetic

This paper considers a class C(Z_p) of closed sets of the p-adic integers obtained by graph-directed constructions analogous to those of Mauldin and Williams over the real numbers. These sets are characterized as collections of those p-adic integers whose p-adic expansions are describeed by paths in the graph of a finite automaton issuing from a distinguished initial vertex. This paper shows that this class of sets is closed under the arithmetic operations of addition and multiplication by p-integral rational numbers. In addition the Minkowski sum (under p-adic addition) of two set in the class is shown to also belong to this class. These results represent purely p-adic phenomena in that analogous closure properties do not hold over the real numbers. We also show the existence of computable formulas for the Hausdorff dimensions of such sets.

math.MG↗

arXiv2013

Euler's constant: Euler's work and modern developments

This paper has two parts. The first part surveys Euler's work on the constant gamma=0.57721... bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part describes various mathematical developments involving Euler's constant, as well as another constant, the Euler-Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations and random matrix products. It includes recent results on Diophantine approximation and transcendence related to Euler's constant.

math.NT↗

arXiv2013

Intersections of multiplicative translates of 3-adic Cantor sets

Motivated by a question of Erdős, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3-adic integers whose forward orbits under this action intersects the 3-adic Cantor set infinitely many times. It has been shown that this set has Hausdorff dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these sets leads to study of intersections of multiplicative translates of Cantor sets by powers of 2. More generally, this paper studies the structure of finite intersections of general multiplicative translates of the 3-adic Cantor set by integers 1 < M_1 < M_2 < ...< M_n. These sets are describable as sets of 3-adic integers whose 3-adic expansions have one-sided symbolic dynamics given by a finite automaton. As a consequence, the Hausdorff dimension of such a set is always of the form log(β) for an algebraic integer β. This paper gives a method to determine the automaton for given data (M_1, ..., M_n). Experimental results indicate that the Hausdorff dimension of such sets depends in a very complicated way on the integers M_1,...,M_n.

math.NT↗

arXiv2013

Ergodicity of the action of K* on A_K

Connes gave a spectral interpretation of the critical zeros of zeta- and L-functions for a global field K using a space of square integrable functions on the space A_K/K* of adele classes. It is known that for K=Q the space A_K/K* cannot be understood classically, or in other words, the action of Q* on A_Q is ergodic. We prove that the same is true for any global field K, in both the number field and function field cases.

math.NT↗