On the sum of reciprocals of least common multiples

Let ${\{a_i\}}_{i=1}^{\infty }$ be a strictly increasing sequence of positive integers ($a_i<a_j$ if $i<j$). In 1978, Borwein showed that for any positive integer n, we have ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,a_{i+1})}\le 1?\frac{1}{2^n}$, with equality occurring if and only if $a_i=2^{i?1}$ for $1\le i\le n+1$. Let $3\le r\le 7$ be an integer. In this paper, we investigate the sum ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}$ and show that ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}\le U_r(n)$ for any positive integer n, where $U_r(n)$ is a constant depending on r and n. Further, for any integer $n\ge 2$, we also give a characterization of the sequence ${\{a_i\}}_{i=1}^{\infty }$ such that the equality ${\sum }_{i=1}^n\frac{1}{\mathrm{lcm}(a_i,...,a_{i+r?1})}=U_r(n)$ holds.     

Algorithm for a sum of reciprocals

An algorithm is described for the approximate calculation of a collection of sums of the form _k= _j–1 ⁿ c_j/(_j+_k), 1kn, where 0<_j. The working time of the algorithm is 0(n(t+ log n)(t+log n)) if _k calculated to within 2^–t; here the function (l) denotes the time of multiplication of twoZ-bit numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 137, pp. 3–6, 1984.In conclusion, the author expresses thanks to A. O. Slisenko and Yu. A. Kuznetsov, who drew the attention of the author to the problem considered.     

On geometry of numbers and uniform rational approximation to the Veronese curve

Periodica Mathematica Hungarica - Consider the classical problem of rational simultaneous approximation to a point in $${mathbb {R}}^{n}$$ . The optimal lower bound on the gap between the induced...     

On the sum of the reciprocals of the differences between consecutive primes

Let \(p_n\) denote the n-th prime number, and let \(d_n=p_{n+1}-p_{n}\). Under the Hardy–Littlewood prime-pair conjecture, we prove

$$\begin{aligned} \sum _{n\le X}\frac{\log ^{\alpha }d_n}{d_n}\sim {\left\{ \begin{array}{ll} \quad \frac{X\log \log \log X}{\log X}~\qquad \quad ~ &{}\alpha =-1,\\ \frac{X}{\log X}\frac{(\log \log X)^{1+\alpha }}{1+\alpha }\qquad &{}\alpha >-1, \end{array}\right. } \end{aligned}$$

and establish asymptotic properties for some series of \(d_n\) without the Hardy–Littlewood prime-pair conjecture.

     

On the sum of the reciprocals of cycle lengths in sparse graphs

For a graphG let ℒ(G)=Σ{1/k contains a cycle of lengthk}. Erdős and Hajnal [1] introduced the real functionf(α)=inf {ℒ (G)|E(G)|/|V(G)|≧α} and suggested to study its properties. Obviouslyf(1)=0. We provef (k+1/k)≧(300k logk)⁻¹ for all sufficiently largek, showing that sparse graphs of large girth must contain many cycles of different lengths.     

On the product sets of rational numbers

A new lower bound on the size of product sets of rational numbers is obtained. An upper estimate for the multiplicative energy of two sets of rational numbers is also found.     

On approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces

This paper discusses the approximation by reciprocals of polynomials with positive coefficients in Orlicz spaces and proved that if f(x) ∈ LM*[0,1], changes its sign at most once in (0,1), then there exists x0 ∈ (0,1) and a polynomial Pn ∈ Πn(+) such that f (x) -Pn (x)x-x0 M ≤ Cω( f,n-1/2 )M, where Πn(+) indicates the set of all polynomials of degree n with positive coefficients.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

On the existence of the best discrete approximation in norm by reciprocals of real polynomials

For the given data (w_i,x_i,y_i), i=1,…,M, we consider the problem of existence of the best discrete approximation in l_p norm (1≤p<∞) by reciprocals of real polynomials. For this problem, the existence of best approximations is not always guaranteed. In this paper, we give a condition on data which is necessary and sufficient for the existence of the best approximation in l_p norm. This condition is theoretical in nature. We apply it to obtain several other existence theorems very useful in practice. Some illustrative examples are also included.     

On estimates of approximation numbers and best bilinear approximation

We obtain estimates of approximation numbers of integral operators, with the kernels belonging to Sobolev classes or classes of functions with bounded mixed derivatives. Along with the estimates of approximation numbers, we also obtain estimates of best bilinear approximation of such kernels.Communicated by Charles A. Micchelli.     

On the normal approximation of a binomial random sum

We present upper bounds of the L _s norms of the normal approximation for random sums of independent identically distributed random variables X ₁ , X ₂ , . . . with finite absolute moments of order 2 + δ, 0 < δ ≤ 1, where the number of summands N is a binomial random variable independent of the summands X ₁ , X ₂ , . . . . The upper bounds obtained are of order (E N) ^?δ/2 for all 1 ≤ s ≤ ∞.     

On Müntz rational approximation

One of the main results of the present paper shows that, for any sequence of real numbers {λ_n} with infinitely many distinct elements, the monotone rational combinations of {X^λn} always form a dense set in the uniform norm in the subspace of monotone functions fromC _{[0, 1]}.     

On discrete rational least squares approximation

Summary The paper deals with the finite rational least squares approximation to a discrete function. An approximation without poles and depending on a parameter is defined which tends to the least squares approximation for 0. It gives an acceptable approximation when the least squares approximation does not exist. Further it is shown that, if the discrete function to be fitted is sufficiently close to a rational function, then the least squares approximation exists.