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In Shallit’s problem (SIAM Review, 1994), it was proposed to justify a two-term asymptotics of the minimum of a rational function of $$n$$ variables defined as the sum of a special form whose number of terms is of order $$n^2$$ as $$n\to\infty$$ . Of particular interest is the second term of this asymptotics (“Shallit’s constant”). The solution published in SIAM Review presented an iteration algorithm for calculating this constant, which contained some auxiliary sequences with certain properties of monotonicity. However, a rigorous justification of the properties, necessary to assert the convergence of the iteration process, was replaced by a reference to numerical data. In the present paper, the gaps in the proof are filled on the basis of an analysis of the trajectories of a two-dimensional dynamical system with discrete time corresponding to the minimum points of $$n$$ -sums. In addition, a sharp exponential estimate of the remainder in Shallit’s asymptotic formula is obtained.

The paper is devoted to the application of the circle method to the problem of an asymptotics of the weighted number of integer points on multidimensional hyperboloids of a special form. We prove the convergence and positivity of the singular series and obtain an asymptotic formula for the singular integral of this problem. Earlier, only estimates for the singular integral were known.

An asymptotic formula for the conformal module of a doubly connected domain symmetric about the abscissa axis under an unbounded stretching in the direction of this axis is obtained. Thereby, in the case of symmetric domains, a question asked by Vuorinen is answered.

An asymptotic formula in the generalized Estermann ternary problem for noninteger powers with almost equal summands dealing with the representation of a sufficiently large natural number as the sum of two primes and the integer part of a noninteger power of a natural number is proved.

For the number of solutions of the congruence xy ≡ 1 (modp) under the graph of a linear function, an asymptotic formula with square root cancellation in the error term is proved.

It is shown that the first two terms in the asymptotic formula for the capacity of a generalized condenser are independent of the shape of degenerating plates. An earlier formula was given only for the case in which the degenerating plates are elements of families of almost disks with fixed centers.

These quantities can be estimated by using estimates for the probabilities of large deviations, i.e., for the quantities pn = λ{|Anf − f∗| ≥ e} (see [1], [2]). It turns out that if, for some essentially bounded function f not only the upper bound, but also the exact exponential asymptotic formula for the probability of large deviations is known, then the same formula is valid also for the rate of convergence in the Birkhoff theorem (see the theorem below). As an example, we shall apply this result to transitive Anosov diffeomorphisms. Suppose that f ∈ L∞(Ω) andΔ = ‖f − f∗‖∞. We assume that the function f ≡ f∗ a.e.; otherwise, the estimate is trivial: Pn = 0 for all n ∈ N and e > 0. For all real e > 0 and d > 1, we set

We prove an asymptotic formula for the number of representations of a sufficiently large natural number N as the sum of two primes p1 and p2 and the cube of a natural numbermsatisfying the conditions |pi − N/3| ≤ H, |m3 − N/3| ≤ H, H ≥ N5/6ℒ10.

We obtain an asymptotic formula for the mean value of the function τk(n), which is the number of solutions of the equation x1 … xk = n in natural numbers x1, …, xk, in some special sequences of natural numbers.

We study the ray propagation path in a plane smoothly irregular waveguide. The following two modes of ray propagation are possible: with reflections and without reflections from the waveguide walls. In each of these modes, the problem has an adiabatic invariant. We obtain an asymptotic formula for the value of the adiabatic invariant jump as the propagation mode changes.

We present sufficient conditions for the existence of an eigenvalue of the Laplace operator with zero Dirichlet conditions in a weakly perturbed infinite cylinder in the case of localized perturbations which are extensions along the transverse coordinates with coefficients depending on the longitudinal coordinate. If such an eigenvalue exists, then, for this eigenvalue, we obtain an asymptotic formula with respect to a small parameter characterizing the values of extensions.

Suppose that the Riemann hypothesis holds. Suppose that $$\psi _1 (x) = \mathop \sum \limits_{\begin{array}{*{20}c} {n \leqslant x} \\ {\{ (1/2)n^{1/c} \} < 1/2} \\ \end{array} } \Lambda (n)$$ where c is a real number, 1 N 1/2+10e, e > 0, the following asymptotic formula is valid: $$\psi _1 (N + H) - \psi _1 (N) = \frac{H}{2}\left( {1 + O\left( {\frac{1}{{N^\varepsilon }}} \right)} \right)$$ .

In this paper, we derive sufficient conditions for the existence of an eigenvalue for the Laplace and the Schrodinger operators with transversal potential for homogeneous Dirichlet boundary conditions in a tube, i.e., in a curved and twisted infinite cylinder. For tubes with small curvature and small internal torsion, we derive an asymptotic formula for the eigenvalue of the problem. We show that, under certain relations between the curvature and the internal torsion of the tube, the above operators possess no discrete spectrum.

We prove an asymptotic formula for the number of representations of a sufficiently large positive integer N as the sum of two primes and the square of a natural number when they are almost equal.

In this paper, we prove the Plancherel--Rotach asymptotic formula for the Chebyshev--Hermite functions $$( - 1)^n e^{x^2 /2} (e^{ - x^2 } )^{(n)} /\sqrt {2^n n!\sqrt \pi } $$ and their derivatives for the case in which $$ + \infty $$ belongs to the domain of definition. A method for calculating the approximation accuracy is also given.

Suppose thatB R d is a ball of radiusR in ℂ d and σ is the standard measure on the unit sphere in ℂ d . ForR>1, 1≤p≤∞, and for the natural numbersl, d, byH R 0 (l, p, d) we denote the class of functionsf holomorphic inB R d and such that in the homogeneous polynomial expansion of the firstl summands the zero and radial derivatives of orderl belong to the closed unit ball of the Hardy spaceH p (B R d ). In this paper an asymptotic formula for the e-entropy of the classH R 0 (l, p, d) in the spacesL p (σ), 1≤p

The Gibbs canonical distribution for a system of N classical particles is studied under the following conditions: the external potential isO(1), the potential of pairwise interaction isO(1/N), the potential of triple interaction isO(1/N 2), etc. The asymptotics of free energy and of the partition function asN→∞ is found. An asymptotic formula approximating the normalized canonical distribution in theL 1 norm asN→∞ is also constructed. It is proved that the chaos property is satisfied fork-particle distributions,k = const, and is not satisfied for theN-particle distribution.

In the paper, an estimate of the number of summands in the asymptotic formula for the number of solutions to Waring's equation is obtained. This is achieved by means of a recurrent process leading to a greater reduction than that in Vinogradov's mean value theorem.

An asymptotic formula for the number of points of an arbitrary lattice in the hyperbolic cross is obtained.

Suppose that 0