Vol 299, No 1 (2017)

An overview is given of the scientific results obtained by Anatolii Alekseevich Karatsuba between the early 1990s and 2008.

New estimates are obtained for complete arithmetic sums of polynomial values (exponential sums, sums of Dirichlet characters, and sums of Bernoulli polynomials) in the case where the derivative of the polynomial in the argument of the sum has no multiple roots modulo primes dividing the period of these arithmetic sums.

Several asymptotic expansions and formulas for cubic exponential sums are derived. The expansions are most useful when the cubic coefficient is in a restricted range. This generalizes previous results in the quadratic case and helps to clarify how to numerically approximate cubic exponential sums and how to obtain upper bounds for them in some cases.

A formula of Atkinson type for the primitive of Hardy’s function is generalized to the case where the lengths of the two sums involved in that formula vary in wide ranges.

A sharpened lower bound is obtained for the number of solutions to an inequality of the form α ≤ {(an̅ + bn)/q} < β, 1 ≤ n ≤ N, where q is a sufficiently large prime number, a and b are integers with (ab, q) = 1, nn̅ ≡ 1 (mod q), and 0 ≤ α < β ≤ 1. The length N of the range of the variable n is of order q^ε, where ε > 0 is an arbitrarily small fixed number.

Haas–Molnar maps are a family of maps of the unit interval introduced by A. Haas and D. Molnar. They include the regular continued fraction map and A. Renyi’s backward continued fraction map as important special cases. As shown by Haas and Molnar, it is possible to extend the theory of metric diophantine approximation, already well developed for the Gauss continued fraction map, to the class of Haas–Molnar maps. In particular, for a real number x, if (p_n/q_n)_n≥1 denotes its sequence of regular continued fraction convergents, set θ_n(x) = q_n²|x − p_n/q_n|, n = 1, 2.... The metric behaviour of the Cesàro averages of the sequence (θ_n(x))_n≥1 has been studied by a number of authors. Haas and Molnar have extended this study to the analogues of the sequence (θ_n(x))_n≥1 for the Haas–Molnar family of continued fraction expansions. In this paper we extend the study of \(({\theta _{{k_n}}}(x))\)_n≥1 for certain sequences (k_n)_n≥1, initiated by the second named author, to Haas–Molnar maps.

In our previous papers, within the theory of the Riemann zeta-function we have introduced the following notions: Jacob’s ladders, oscillating systems, ζ-factorization, metamorphoses, etc. In this paper we obtain a ζ-analogue of an elementary trigonometric identity and other interactions between oscillating systems.

For a nonprincipal character χ modulo D, we prove a nontrivial estimate of the form Σ_n≤x Λ(n)χ(n − l) \( \ll x\exp \{ - 0.6\sqrt {\ln D} \} \) for the sum of values of χ over a sequence of shifted primes in the case when x ≥ D^1/2+ε, (l,D) = 1, and the modulus of the primitive character generated by χ is a cube-free number.

A simplex–karyon algorithm for expanding real numbers α = (α₁,..., α_d) in multidimensional continued fractions is considered. The algorithm is based on a (d + 1)-dimensional superspace S with embedded hyperplanes: a karyon hyperplane K and a Farey hyperplane F. The approximation of numbers α by continued fractions is performed on the hyperplane F, and the degree of approximation is controlled on the hyperplane K. A local ℘(r)-strategy for constructing convergents is chosen, with a free objective function ℘(r) on the hyperplane K.