Benford's law for the $3x+1$ function
We show that for most choices of an initial seed $x_0$, the sequence of the first $N$ iterates of $x_0$ under the $3x+1$ map approximately satisfies Benford's law.
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arXiv subjects
Jeffrey C. Lagarias
Publications and source records attributed to Jeffrey C. Lagarias.
A Note on Absolute Derivations and Zeta Functions
This comment answers a question raised by Kurokawa, Ochiai and Wakayama whether a certain operator constructed using a notion of quantum non-commutativity of primes has eigenvalues related to the Riemann zeta zeros.
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Zero Spacing Distributions for Differenced L-Functions
This paper studies the local spacings of deformations of the Riemann zeta function under certain averaging and differencing operations. For real h it considers A_h(s)= 1/2(xi(s+h)+ xi(s-h)) and B_h(s)=1/(2i)(xi(s+h)-xi(s-h)), where xi(s) is the Riemann xi-function. For |h|\ge 1/2 all zeros of these functions lie on the critical line and are simple zeros. Assuming the Riemann hypothesis, the same holds for all nonzero h. The number of zeros to height T has the same asymptotics as for the zeta function in its main terms, so one can define normalized zero spacings as for the xi-function. It is shown that the first k normalized zero spacings have a limiting distribution as T goes to infinity, and it is the "trivial" distribution with all spacings equal to 1. This is shown unconditionally for |h|> 1/2 and conditionally on RH for 0< |h|< 1/2. Similar results are indicated for Dirichlet L-functions. The results are interpreted in terms of de Branges's theory of Hilbert spaces of entire functions.
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The Riemann hypothesis for certain integrals of Eisenstein series
This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical line Re(s) = 1/2. For the constant term of the Eisenstein series it shows that all zeros are on the critical line for fixed y= Im(z) \ge 1, except possibly for two real zeros, which are present if and only if y > 4 πe^{-γ} = 7.0555+. It shows the Riemann hypothesis holds for all truncation integrals with truncation parameter T \ge 1. For T=1 this proves the Riemann hypothesis for a zeta function recently introduced by Lin Weng, attached to rank 2 semistable lattices over the rationals.
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Dynamics of a family of piecewise-linear area-preserving plane maps II. Invariant circles
This paper studies the behavior under iteration of the maps T_{ab}(x,y)=(F_{ab}(x)-y,x) of the plane R^2, in which F_{ab}(x)=ax if x>=0 and bx if x<0. The orbits under iteration correspond to solutions of the nonlinear difference equation x_{n+2}= 1/2(a-b)|x_{n+1}| + 1/2(a+b)x_{n+1} - x_n. This family of maps has the parameter space (a,b)\in R^2. These maps are area-preserving homeomorphisms of R^s that map rays from the origin into rays from the origin. This paper shows the existence of special parameter values where T_{ab} has every nonzero orbit an invariant circle with irrational rotation number, and these invariant circles are piecewise unions of arcs of conic sections. Numerical experiments suggest the possible existence of many other parameter values having invariant circles.
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Li Coefficients for Automorphic L-Functions
Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients lambda_n, indexed by the integers. We define similar coefficients attached to principal automorphic L-functions over GL(N). We relate these coefficients to values of Weil's quadratic functional for the associated automorphic representation, and deduce a Riemann hypothesis criterion in terms of positivity of the real parts of these coefficients. We determine asymptotics of the coefficients, both unconditionally and on the Riemann hypothesis, as n increases. We show the existence of an entire function of exponential type that interpolates the generalized Li coefficients at integer values.
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The 3x+1 Semigroup
The 3x+1 semigroup is the multiplicative semigroup generated by the rational numbers of form (2k+1)/(3k+2) for non-negative k, together with 2. This semigroup encodes backward iteration under the 3x+1 map, and the 3x+1 conjecture implies that it contains every positive integer. We prove this is the case, and show that this semigroup consists of all positive rational numbers a/b such that 3 does not divide b.
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Wild and Wooley Numbers
This paper studies the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n, together with 1/2. The subsemigroup of integers in this semigroup is called the wild integer semigroup, and the wild numbers are the irreducible elements in this subsemigroup. This paper presents evidence that the wild numbers consist of all the prime numbers p except 3. The subsemigroup of integers in the multiplicative semigroup generated by all rationals of the form (3n+2)/(2n+1) for nonnegative integers n is called the Wooley integer semigroup and its irreducible elements are called Wooley numbers. This semigroup is shown to be recursive, and various open problems are formulated about Wooley integers.
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Apollonian Circle Packings: Number Theory II. Spherical and Hyperbolic Packings
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. In Euclidean space it is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. There are infinitely many different integral packings; these were studied in the paper \cite{GLMWY21}. Integral circle packings also exist in spherical and hyperbolic space, provided a suitable definition of curvature is used (see \cite{LMW02}) and again there are an infinite number of different integral packings. This paper studies number-theoretic properties of such packings. This amounts to studying the orbits of a particular subgroup $\sA$ of the group of integral automorphs of the indefinite quaternary quadratic form $Q_{\sD}(w, x, y, z)= 2(w^2+x^2 +y^2 + z^2) - (w+x+y+z)^2$. This subgroup, called the Apollonian group, acts on integer solutions $Q_{\sD}(w, x, y, z)=k$. This paper gives a reduction theory for orbits of $\sA$ acting on integer solutions to $Q_{\sD}(w, x, y, z)=k$ valid for all integer $k$. It also classifies orbits for all $k \equiv 0 \pmod{4}$ in terms of an extra parameter $n$ and an auxiliary class group (depending on $n$ and $k$), and studies congruence conditions on integers in a given orbit.
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Area Inequalities for Embedded Disks Spanning Unknotted Curves
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r^2. In the direction of lower bounds, we give a sequence of length one curves with r approaching 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A.
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The Number of Triangles Needed to Span a Polygon Embedded in R^d
Given a closed polygon P having n edges, embedded in R^d, we give upper and lower bounds for the minimal number of triangles t needed to form a triangulated PL surface in R^d having P as its geometric boundary. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert suface construction to show there always exists an embedded surface requiring at most 7n^2 triangles. We complement this result by showing there are polygons in R^3 for which any embedded surface requires at least 1/2n^2 - O(n) triangles. In dimension 2 only n-2 triangles are needed, and in dimensions 5 or more there exists an embedded surface requiring at most n triangles. In dimension 4 we obtain a partial answer, with an O(n^2) upper bound for embedded surfaces, and a construction of an immersed disk requiring at most 3n triangles. These results can be interpreted as giving qualitiative discrete analogues of the isoperimetric inequality for piecewise linear manifolds.
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Substitution Delone Sets
This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.
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On the Normality of Arithmetical Constants
D. Bailey and R. E. Crandall recently formulated a "Hypothesis A", which provides a general principle to explain the (conjectured) normality of constants like pi or log 2 and other related numbers, to base 2 or other integer bases. This paper explains the basic mechanism behind their principle as a connection between single orbits of two different discrete dynamical systems. It relates a subclass of arithmetical constants they consider to special values of G-functions, and characterizes this subclass. Finally it notes some parallels of "Hypothesis A" with Furstenberg's conjecture on invariant measures.
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On a two-variable zeta function for number fields
This paper studies a zeta function of two complex variables (w, s) attached to an algebraic number field K, introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. We mainly consider the case of the rational field Q, where for w = 1 one recovers the Riemann zeta function with the factors at infinity added. The analogue of the Riemann xi-function analytically continues to an entire function of two complex variables, and satisfies a functional equation holding w fixed and sending s to w-s. For real w the "critical line" is therefore Re(s) = w/2. For fixed nonnegative real w the zeros are confined to a vertical strip in s and have the same asymptotics as zeta zeros. For negative real w the function is positive real on the critical line, so has no zeros there. This phenomonon is associated to a positive convolution semigroup of infinitely divisible probability distributions, and the Khintchine canonical measure of this family is explicitly determined.
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Bounds for the 3x+1 Problem using Difference Inequalities
We study difference inequality systems for the 3x+1 problem introduced by the first author in 1989. These systemes can be used to give lower bounds for the number of integers below x that contain 1 in their forward orbit under the 3x+1 map. Previous methods gave away some information in these inequalities. We give an improvement which apparantly extracts full information from the inequalities. By computer aided proof we show that at least x^{0.84} of the integers below x contain 1 in their forward orbit under the 3x+1 map.
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Affine isoperimetric inequalities for piecewise linear surfaces
This paper considers affine analogues of the isoperimetric inequality in the sense of piecewise linear topology. Given a closed polygon P embedded in R^d having n edges, we give upper and lower bounds for the minimal number of triangles needed to forma triangulated embedded orientable surface in R^d having P as its geometric boundary. The most interesting case is dimension dimension 3, where we give an upper bound of 7 n^2 triangles, and a lower bound for some polygons P that require at least 1/2 n^2 triangles. In dimension 2 and dimensions 5 and above one needs only O(n) triangles. The case of dimension 4 is not completely resolved.
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The Kruskal Count
The Kruskal Count is a card trick invented by Martin J. Kruskal in which a magician "guesses" a card selected by a subject according to a certain counting procedure. With high probability the magician can correctly "guess" the card. The success of the trick is based on a mathematical principle related to coupling methods for Markov chains. This paper analyzes in detail two simplified variants of the trick and estimates the probability of success. The model predictions are compared with simulation data for several variants of the actual trick.
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An Elementary Problem Equivalent to the Riemann Hypothesis
This paper shows the equivalence of the Riemann hypothesis to an sequence of elementary inequalities involving the harmonic numbers H_n, the sum of the reciprocals of the integers from 1 to n. It is a modification of a criterion due to Guy Robin.
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