ABSTRACT We describe connections between the de Branges theory of Hilbert spaces of entire functions and the Riemann hypothesis for Dirichlet L-functions. Assuming the Riemann hypothesis holds for a given L-function, there exists an... more
ABSTRACT We describe connections between the de Branges theory of Hilbert spaces of entire functions and the Riemann hypothesis for Dirichlet L-functions. Assuming the Riemann hypothesis holds for a given L-function, there exists an associated de Branges space with interesting properties, and conversely. This de Branges space comes with an associated self-adjoint operator having as eigenvalues the imaginary parts of the L-function zeros on the critical line, and this operator has an interpretation as a “Hilbert-Polya” generalized differential operator.
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Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is... more
Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.
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We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.
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Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is... more
Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic on the unit disk $|z|<1$ and analytic on $|z|<1/\beta$. This paper shows that the singularities of $L_{\beta,\alpha}(z)$ on the circle $|z|=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N):0\le l\le N-1\}$, for some integer $N\ge 1$. Here $N$ can be taken to be the period $N_{\beta,\alpha}$ of a certain Markov chain $\Sigma_{\beta,\alpha}$ which encodes information about generalized lap numbers $L_{n}(i,j)$ of $f_{\beta,\alpha}$, where $L_{n}(i,j)$ counts monotonic pieces of $f_{\beta,\alpha}^{n}$ whose image is $[f^{i}(0),f^{j}(1^{-}))$. We show that $N_{\beta,\alpha}=1$ whenever $\beta>2$. Finally, we give the criterion that $N_{\beta,\alpha}=1$ if and only if for all $n\ge 1$ the map $f_{\beta,\alpha}^{n}$ is ergodic with respect to the maximal entropy measure of $f_{\beta,\alpha}$.
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Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is... more
Linear mod one transformations are those maps of the unit interval given by $f_{\beta,\alpha}(x)=\beta x+\alpha$ (mod 1), with $\beta>1$ and $0\le\alpha<1$. The lap-counting function is $L_{\beta,\alpha}(z)=\sum_{n=1}^{\infty} L_{n}z^{n}$, where $L_{n}$ essentially counts the number of monotonic pieces of the $n$th iterate $f_{\beta,\alpha}^{n}$. Part I showed that the function $L_{\beta,\alpha}(z)$ is meromorphic in the unit disk $\vert z\vert <1$ and analytic in $\vert z\vert<1/\beta$, and part II showed that the singularities of $L_{\beta,\alpha}(z)$ on the circle $\vert z\vert=1/\beta$ are contained in the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$, where $N_{\beta,\alpha}$ is the period of the ergodic part of a Markov chain associated to $f_{\beta,\alpha}$. This paper proves that the set of singularities on $\vert z\vert=1/\beta$ is identical to the set $\{(1/\beta)\exp (2\pi il/N_{\beta,\alpha}):0\le l/N_{\beta,\alpha}\}$. Part II showed that $N_{\beta,\alpha}=1$ for $\beta> 2$, and this paper determines $N_{\beta,\alpha}$ in the remaining cases where $1<\beta\le 2$.
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Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of... more
Linear mod one transformations are the maps of the unit interval given by fβα(x) = βx + α (mod 1), with β > 1 and 0 ≤ α < 1. The lap-counting function is the function where the lap number Ln essentially counts the number of monotonic pieces of the nth iterate . We derive an explicit factorization formula for Lβα(z) which directly shows that Lβα(z) is a function meromorphic in the open unit disk {z: |z| < 1} and analytic in the open disk {z: |z| < 1/β}, with a simple pole at z = 1/β.Comparison with a known formula for the Artin—Mazur—Ruelle zeta function ζβ,α(z) of fβα shows that Lβα(z) and ζβ,α(z) have identical sets of singularities in the disk {z: |z| < 1}. We derive two more factorization formulae for Lβ,α(z) valid for certain parameter ranges of (β, α). When 1 < α + β ≤ 2, there is sometimes a ‘renormalization’ structure of such maps present, which has previously been studied in connection with simplified models for the Lorenz attractor. In the case that fβα is...
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A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral... more
A lattice polytope is a polytope in whose vertices are all in . The volume of a lattice polytope P containing exactly k ≥ 1 points in d in its interior is bounded above by . Any lattice polytope in of volume V can after an integral unimodular transformation be contained in a lattice cube having side length at most n˙n ! V. Thus the number of equivalence classes under integer unimodular transformations of lattice poly topes of bounded volume is finite. If S is any simplex of maximum volume inside a closed bounded convex body K in having nonempty interior, then K⊆ ( n + 2)S — (n+ l)s where mS denotes a nomothetic copy of S with scale factor m, and s is the centroid of S.
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P. Erdős asked how frequently does 2n have a ternary expansion that omits the digit 2. He conjectured that this holds only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems.... more
P. Erdős asked how frequently does 2n have a ternary expansion that omits the digit 2. He conjectured that this holds only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first considers truncated ternary expansions of real sequences xn(λ) = ⌊λ2n⌋, where λ > 0 is a real number, along with its untruncated version, while the second considers 3-adic expansions of sequences yn(λ) = λ2 n, where λ is a 3-adic integer. We show in both cases that the set of initial values having infinitely many iterates that omit the digit 2 is small in a suitable sense. For each nonzero initial value we obtain an asymptotic upper bound as k → ∞ on the the number of the first k iterates that omit the digit 2. We also study auxiliary problems concerning the Hausdorff dimension of intersections of multiplicative translates of 3-adic Cantor sets.
Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of the set of coefficients λn = ∑ ρ 1 − ( 1 − 1ρ n , (n = 1, 2, ...), in which ρ runs over the nontrivial zeros of the Riemann zeta function. We define... more
Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of the set of coefficients λn = ∑ ρ 1 − ( 1 − 1ρ n , (n = 1, 2, ...), in which ρ runs over the nontrivial zeros of the Riemann zeta function. We define similar coefficients λn(π) associated to principal automorphic L-functions L(s, π) over GL(N). We relate these cofficients to values of Weil’s quadratic functional associated to the representation π on a suitable set of test functions. The positivity of the real parts of these coefficients is a necessary and sufficient condition for the Riemann hypothesis for L(s, π). We derive an unconditional asymptotic formula for the coefficients λn(π), in terms of the zeros of L(s, π). Assuming the Riemann hypothesis for L(s, π), we deduce that λn(π) = N 2 n log n + C1(π)n + O( √ n log n), where C1(π) is a real-valued constant and the implied constant in the remainder term depends on π. We also show that there exists a entire function Fπ(z) of exponential type tha...
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The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between... more
The Alternative Hypothesis (AH) concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced, which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between non-trivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$, which satisfy the Alternative Hypothesis spacing, but mimic the local statistics that are currently known a...
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The 3x + 1 problem concerns iteration of the map T : Z! Zgiven by T(x) = 3x + 1 2 if x 1 (mod 2) : x 2 if x 0 (mod 2) : The 3x + 1 Conjecture asserts that each m 1 has some iterate T (k) (m) = 1. This is an annotated bibliography of work... more
The 3x + 1 problem concerns iteration of the map T : Z! Zgiven by T(x) = 3x + 1 2 if x 1 (mod 2) : x 2 if x 0 (mod 2) : The 3x + 1 Conjecture asserts that each m 1 has some iterate T (k) (m) = 1. This is an annotated bibliography of work done on the 3x + 1 problem and related problems subsequent to the survey of Lagarias (1985).
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Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such... more
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. 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. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: 2(x2 + y2 + z2 + w2) − (x + y + z + w)2 = 0. Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We determine asymptotics for the number of root quadruples of size below T . We study which integers occur in a given integer packing, and determine con...
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such... more
Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. 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. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: x2 + y2 + z2 + w2 = 12(x + y + z + w) 2. Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element −n as a class number, and give an exact formula for it. We study which integers occur in...
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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 embedded in ℝ d having P as its geometric boundary. More generally we... more
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 embedded in ℝ d having P as its geometric boundary. More generally we obtain such bounds for a triangulated (locally flat) PL surface having P as its boundary which is immersed in ℝ d and whose interior is disjoint from P. The most interesting case is dimension 3, where the polygon may be knotted. We use the Seifert surface construction to show that for any polygon embedded in ℝ3 there exists an embedded orientable triangulated PL surface having at most 7n 2 triangles, whose boundary is a subdivision of P. We complement this with a construction of families of polygons with n vertices for which any such embedded surface requires at least 12n 2−O(n) triangles. We also exhibit families of polygons in ℝ3 for which Ω(n 2) triangles are required in any immersed PL surface of the above kind. In contrast, in dimension 2 and in dimensions d ≥ 5 there always exists an embedded locally flat PL disk having P as boundary that contains at most n triangles. In dimension 4 there always exists an immersed locally flat PL disk of the above kind that contains at most 3n triangles. An unresolved case is that of embedded PL surfaces in dimension 4, where we establish only an O(n 2) upper bound. These results can be viewed as providing qualitative discrete analogues of the isoperimetric inequality for piecewise linear (PL) manifolds. In dimension 3 they imply that the (asymptotic) discrete isoperimetric constant lies between 1/2 and 7.
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We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem,... more
We consider the problem of deciding whether a polygonal knot in 3-dimensional Euclidean space is unknotted, ie., capable of being continuously deformed without self-intersection so that it lies in a plane. We show that this problem, UNKNOTTING PROBLEM is in NP. We also consider the problem, SPLITTING PROBLEM of determining whether two or more such polygons can be split, or continuously deformed without self-intersection so that they occupy both sides of a plane without intersecting it. We show that it also is in NP. Finally, we show that the problem of determining the genus of a polygonal knot (a generalization of the problem of determining whether it is unknotted) is in PSPACE. We also give exponential worst-case running time bounds for deterministic algorithms to solve each of these problems. These algorithms are based on the use of normal surfaces and decision procedures due to W. Haken, with recent extensions by W. Jaco and J. L. Tollefson.