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Register a new userOn the difference between a D. H. Lehmer number and its inverse over short interval
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Let $q>2$ be an odd integer. For each integer $x$ with $0<x<q$ and $(q,x)= 1$, we know that there exists one and only one $bar{x}$ with $0<bar{x}<q$ such that $xbar{x}equiv1(bmod q)$. A Lehmer number is defined to be any integer $a$ with $2dagger(a+bar{a})$. For any nonnegative integer $k$, Let $$ M(x,q,k)=displaystylemathop {displaystylemathop{sum{}}_{a=1}^{q} displaystylemathop{sum{}}_{bleq xq}}_{mbox{$tinybegin{array}{c} 2|a+b+1 abequiv1(bmod q)end{array}$}}(a-b)^{2k}.$$ The main purpose of this paper is to study the properties of $M(x,q,k)$, and give a sharp asymptotic formula, by using estimates of Kloostermans sums and properties of trigonometric sums.
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For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(psi)$ of complex numbers which are well approximated with the given bound $psi$ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound $psi$ and its Hausdorff dimension is equal to that of the $psi$-approximable set $W(psi)$ of complex numbers. As a consequence, we also obtain an analogue of the classical Jarnik Theorem in real case.
Let $K$ be a 1-dimensional function field over an algebraically closed field of characteristic $0$, and let $A/K$ be an abelian surface. Under mild assumptions, we prove a Lehmer-type lower bound for points in $A(bar{K})$. More precisely, we prove that there are constants $C_1,C_2>0$ such that the normalized Bernoulli-part of the canonical height is bounded below by $$ hat{h}_A^{mathbb{B}}(P) ge C_1bigl[K(P):Kbigr]^{-2} $$ for all points $Pin{A(bar{K})}$ whose height satisfies $0<hat{h}_A(P)le{C_2}$.
We construct fields of algebraic numbers that have the Lehmer property but not the Bogomolov property. This answers a recent implicit question of Pengo and the first author.
We report the first determination of the relative strong-phase difference between D^0 -> K^0_S,L K^+ K^- and D^0-bar -> K^0_S,L K^+ K^-. In addition, we present updated measurements of the relative strong-phase difference between D^0 -> K^0_S,L pi^+ pi^- and D^0-bar -> K^0_S,L pi^+ pi^-. Both measurements exploit the quantum coherence between a pair of D^0 and D^0-bar mesons produced from psi(3770) decays. The strong-phase differences measured are important for determining the Cabibbo-Kobayashi-Maskawa angle gamma/phi_3 in B^- -> K^- D^0-tilde decays, where D^0-tilde is a D^0 or D^0-bar meson decaying to K^0_S h^+ h^- (h=pi,K), in a manner independent of the model assumed to describe the D^0 -> K^0_S h^+ h^- decay. Using our results, the uncertainty in gamma/phi_3 due to the error on the strong-phase difference is expected to be between 1.7 and 3.9 degrees for an analysis using B^- K^- D^0-tilde D^0-tilde -> K^0_S pi^+ pi^- decays, and between 3.2 and 3.9 degrees for an analysis based on B^- -> K^- D^0-tilde, D^0-tilde -> K^0_S K^+ K^- decays. A measurement is also presented of the CP-odd fraction, F_-, of the decay D^0 -> K^0_S K^+ K^- in the region of the phi -> K^+ K^- resonance. We find that in a region within 0.01 GeV^2/c^4 of the nominal phi mass squared F_- > 0.91 at the 90% confidence level.
Most hypersurfaces in projective space are irreducible, and rather precise estimates are known for the probability that a random hypersurface over a finite field is reducible. This paper considers the parametrization of space curves by the appropriate Chow variety, and provides bounds on the probability that a random curve over a finite field is reducible.
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