Approximation of number π by algebraic numbers from special fields

The estimate from below of the modulus of the difference between π and algebraic numbers from the fields generated by the roots of unity is made.     

On the linear independence of numbers over number fields

In the present paper, the problem of a lower bound for the measure of linear independence of a given collection of numbers ₁, , _n is considered under the assumption that, for a sequence of polynomials whose coefficients are algebraic integers, upper and lower estimates at the point ( ₁, , _n ) are known. We use a method that generalizes the Nesterenko method to the case of an arbitrary algebraic number field.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 506–517, October, 1998.The author wishes to thank Professor Yu. V. Nesterenko for setting the problem and valuable advice and Professor D. Bertrand for fruitful discussions.This research was partially supported by the International Science Foundation (Soros Foundation) under grant No. 507_s.     

On the distribution of numbers of a special form

Translated from Matematicheskie Zametki, Vol. 55, No. 2, pp. 157–161, February, 1994.     

On prime numbers of special kind on short intervals

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 < c ≤ 2. We prove that, for H>N ^1/2+10ε, ε > 0, the following asymptotic formula is valid: $$\psi _1 (N + H) - \psi _1 (N) = \frac{H}{2}\left( {1 + O\left( {\frac{1}{{N^\varepsilon }}} \right)} \right)$$ .     

On equivalent number fields with special Galois groups

In [12] and [13] Jack Sonn has introduced and studied a new notion of equivalence for number fields. In this note we show that “almost all” (cf. [14]) pairs of equivalent number fields are conjugate over ℚ, and we study equivalence classes of fields of prime degree.     

On tame and wild kernels of special number fields

Special number fields are those number fields F for which the wild kernel properly contains the group of divisible elements of K₂(F). We examine the relationship between the wild kernel, the tame kernel and the group of divisible elements for such number fields. For a special number field F we show that WK₂(F)⊂K₂(F)^2b, but WK₂(F)⊄K₂(F)^2b+2 where . We examine analogous questions for instead of K₂(F). As an application, we determine those number fields for which there exist ‘exotic’ Steinberg symbols with values in a finite cyclic group and we show how to construct these exotic symbols.     

Counting the number of twin Niven numbers

Given an integer q≥2, we say that a positive integer is a q-Niven number if it is divisible by the sum of its digits in base q. Given an arbitrary integer r∈[2,2q], we say that (n,n+1,…,n+r−1) is a q-Niven r -tuple if each number n+i, for i=0,1,…,r−1, is a q-Niven number. We show that there exists a positive constant c=c(q,r) such that the number of q-Niven r-tuples whose leading component is <x is asymptotic to cx/(log x) ^r as x→∞. Research of J.M. De Koninck supported in part by a grant from NSERC. Research of I. Kátai supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA.     

Carmichael numbers in number rings

We generalize Carmichael numbers to ideals in number rings and prove a generalization of Korselt's Criterion for these Carmichael ideals. We investigate when Carmichael numbers in the integers generate Carmichael ideals in the algebraic integers of abelian number fields. In particular, we show that given any composite integer n, there exist infinitely many quadratic number fields in which n is not Carmichael. Finally, we show that there are infinitely many abelian number fields K with discriminant relatively prime to n such that n is not Carmichael in K.     

Divergence rates for the number of rare numbers

Suppose thatX ₁,X ₂, ... is a sequence of i.i.d. random variables taking value inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY _n /E[Y _n ] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Y _n ]. These results extend earlier work by the author.     

On the number of edges in induced subgraphs of a special distance graph

We obtain new estimates for the number of edges in induced subgraphs of a special distance graph.     

Maximal class numbers of CM number fields

Fix a totally real number field F of degree at least 2. Under the assumptions of the generalized Riemann hypothesis and Artin's conjecture on the entirety of Artin L-functions, we derive an upper bound (in terms of the discriminant) on the class number of any CM number field with maximal real subfield F. This bound is a refinement of a bound established by Duke in 2001. Under the same hypotheses, we go on to prove that there exist infinitely many CM-extensions of F whose class numbers essentially meet this improved bound and whose Galois groups are as large as possible.     

Class numbers of imaginary abelian number fields

Let be an imaginary abelian number field. We know that , the relative class number of , goes to infinity as , the conductor of , approaches infinity, so that there are only finitely many imaginary abelian number fields with given relative class number. First of all, we have found all imaginary abelian number fields with relative class number one: there are exactly 302 such fields. It is known that there are only finitely many CM-fields with cyclic ideal class groups of 2-power orders such that the complex conjugation is the square of some automorphism of . Second, we have proved in this paper that there are exactly 48 such fields.