Ognian Trifonov

On gaps between squarefree numbers II

A squarefree number is a positive integer not divisible by the square of an integer > 1. We investigate here the problem of finding small h = h(x) such that for x sufficiently large, there is a squarefree number in the interval (x,x + h]. This problem was originally investigated by Fogels [3]; he showed that for every ∈ > 0, h = x 2/5+∈ is admissible. Later Roth [9] reported elementary arguments of Davenport and Estermann showing respectively that one can take h ≫ x 1/3 and h ≫ x 1/3(log x)-2/3 for sufficiently large choices of the implied constants. Roth then gave an elementary proof that h = x 1/4+ ∈ is admissible, and by applying a result of van der Corput, he showed that one can take h≫ x 3/13 (log x)4/13Nair [6] later noted that the elementary proof could be modified to omit the ∈ in the exponent to get that h≫ x 1/4 is admissible, and more recently the first author [1] showed that one could obtain the result h≫x 3/13 by elementary means. Using further exponential sum techniques, Richert [8], Rankin [7], Schmidt [10], and Graham and Kolesnik [4] obtained the improvements h ≫ x 2/9log x, h = x θ +€ where θ = 0.221982…, θ = 109556/494419 = 0.221585…, and θ = 1057/4785 = 0.2208986…, respectively. The authors investigated the problem further.

Order of convergence of second order schemes based on the minmod limiter

Many second order accurate nonoscillatory schemes are based on the minmod limiter, e.g., the Nessyahu-Tadmor scheme. It is well known that the Lp-error of monotone finite difference methods for the linear advection equation is of order 1/2 for initial data in W 1 (Lp), 1 ≤ p < 00. For second or higher order nonoscillatory schemes very little is known because they are nonlinear even for the simple advection equation. In this paper, in the case of a linear advection equation with monotone initial data, it is shown that the order of the L 2 -error for a class of second order schemes based on the minmod limiter is of order at least 5/8 in contrast to the 1/2 order for any formally first order scheme.

On the factorization of consecutive integers

A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, or upon providing lower bounds for the number of prime divisors of such a product. As noted in [18], a basic technique in these arguments is to make a careful distinction between “small” and “large” primes, and then apply sophisticated results from multiplicative number theory. Along these lines, if we write ( n k ) = U · V, n ≥ 2k,

One-sided stability and convergence of the Nessyahu–Tadmor scheme

Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data.

ON VALUES OF d(n!)/m!, ϕ(n!)/m! AND σ(n!)/m!

For f one of the classical arithmetic functions d, ϕ and σ, we establish constraints on the quadruples (n, m, a, b) of integers satisfying f(n!)/m! = a/b. In particular, our results imply that as nm tends to infinity, the number of distinct prime divisors dividing the product of the numerator and denominator of the fraction f(n!)/m!, when reduced, tends to infinity.

On Convergence of Minmod-Type Schemes

A class of nonoscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered. This class of methods contains the classical Lax--Friedrichs and the second-order Nessyahu--Tadmor schemes. In the case of linear flux, new l2 stability results and error estimates for the methods are proved. Numerical experiments confirm that these methods are one-sided l2 stable for convex flux instead of the usual Lip+ stability.

Distribution of interpolation points of bestL2-approximants (nth partial sums of Fourier series)

AbstractsFor a continuous 2π-periodic real-valued functionf, we investigate the asymptotic behavior of the zeros of the errorf(θ)−sn(θ), wheresn(θ) is thenth Fourier section. We prove that there is a subsequence {nk} for which such zeros (interpolation points) are uniformly distributed on [−π, π]. This extends previous results of Saff and Shekhtman. Moreover, results dealing with the maximal distance between consecutive zeros off−snk are obtained. The technique of proof involves coefficient estimates for lacunary trigonometric polynomials in terms of itsLq-norm on a subinterval.

Starting with gaps between k-free numbers

We survey various developments in Number Theory that were inspired by classical papers by Roth [On the gaps between squarefree numbers, J. London Math. Soc. 26 (1951) 263–268] and by Halberstam and Roth [On the gaps between consecutive k-free integres, J. London Math. Soc. 26 (1951) 268–273].

On a problem of Ore

Let kα be the least positive integer such that 2kα is not a value of Euler’s phi-function. In the 1960s P. Bateman and J. Selfridge showed that kα exists for all positive integers α and computed kα for α ≤ 2312. Bateman also formulated a certain conjecture concerning the numbers kα. We show that Bateman’s conjecture does not hold and prove that a modified version of the conjecture holds. Also, let vα be the least positive integer such that 2vα is not a value of the σ function. We show that vα ≤ 509203 for all α, and establish a connection between a certain property of the Mersenne primes and the behavior of the sequence {vα}.