The additive subgroup generated by a polynomial

SupposeR is a prime ring with the centerZ and the extended centroidC. Letp(x ₁, …,x _n) be a polynomial overC in noncommuting variablesx ₁, …,x _n. LetI be a nonzero ideal ofR andA be the additive subgroup ofRC generated by {p(a ₁, …,a _n):a ₁, …,a _n ∈I}. Then eitherp(x ₁, …,x _n) is central valued orA contains a noncentral Lie ideal ofR except in the only one case whereR is the ring of all 2 × 2 matrices over GF(2), the integers mod 2.     

Discrete uniform distribution for a sum of additive functions

We consider the limit distribution of values of a sum of sets of strongly additive arithmetic functions with shifted argument. We obtain sufficient and necessary conditions for a weak convergence of distributions of that sum to the discrete uniform law. The case where those functions take values 0 or 1 on primes is studied.     

Asymptotic normality of additive functions on polynomial sequences in canonical number systems

The objective of this paper is the study of functions which only act on the digits of an expansion. In particular, we are interested in the asymptotic distribution of the values of these functions. The presented result is an extension and generalization of a result of Bassily and Kátai to number systems defined in a quotient ring of the ring of polynomials over the integers.     

代数体函数微分多项式的值分布

利用亚纯函数的Nevanlinna值分布理论和方法,研究了一类代数体函数微分多项式的值分布问题,推广和改进了相关文献的结论.     

Estimate of the concentration function for a class of additive functions

We show how methods for estimating the concentration function of the sum of independent random variables can be used to obtain estimates of the concentration function for the values of additive functions under suitable conditions. Previously obtained estimates of the concentration function are consequences of the estimate obtained in the present paper for functions from of the class under consideration.     

The convergence of distribution of integer-valued additive functions to the poisson law

Letf _x (m), x≥1 be a set of additive functions, and

Nonuniform estimates of the distribution of additive arithmetical functions

Department of Mathematics, Vilnius University. Published in Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 31, No. 4, pp. 584–593, October–December, 1991.     

The digit generating function of a polynomial

The average frequency of 1 occurring as the kth digit in the binary expansion of squares, cubes, and generally the values of a polynomial is studied. In particular, it turns out that the generating function of these frequencies is rational for the important special cases of powers, linear and quadratic polynomials. For higher degree polynomials, the behaviour seems to be much more chaotic in general, which is exhibited by two examples of cubic polynomials.