On the local distribution of certain arithmetic functions

Let d(n), σ ₁(n), and φ(n) stand for the number of positive divisors of n, the sum of the positive divisors of n, and Euler’s function, respectively. For each ν ∈, Z, we obtain asymptotic formulas for the number of integers n ⩽ x for which e _n = 2 ^v r for some odd integer m as well as for the number of integers n ⩽ x for which e _n = 2 ^v r for some odd rational number r. Our method also applies when φ(n) is replaced by σ ₁(n), thus, improving upon an earlier result of Bateman, Erdős, Pomerance, and Straus, according to which the set of integers n such that is an integer is of density 1/2. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 315–331, July–September, 2006.     

On certain arithmetic functions involving the greatest common divisor

The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ²(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.     

On mean values of some arithmetic functions in number fields

Deza and Varukhina [3] established asymptotic formulae for some arithmetic functions in quadratic and cyclotomic fields. We generalize their results to any Galois extension of the rational field. During this process we rectify the main terms in their asymptotic formulae.     

Distribution of values of certain classes of additive arithmetic functions in algebraic number fields

An investigation is made of the generalization of a theorem of B. V. Levin and A. S. Fainleib for homothetically extending regions in a certain n-dimensional real space connected with a given field K of algebraic numbers of degree n2; the paper also investigates applications of the theorem to the problem of the distribution of real additive functions which are given on a set of ideal numbers and which belong to a wider class than the class H of I. P. Kubilyus.Translated from Matematicheskie Zametki, Vol. 4, No. 1, pp. 63–74, July, 1968.     

On mean values of some arithmetic functions in number fields

We obtain, for quadratic and cyclotimic fields, asymptotic formulas for two arithmetic functions, which are similar to divisor function.     

Asymptotic formulas for certain arithmetic functions

This is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere. The last named author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P18079-N12.     

On the distribution of values of certain divisor functions

Let {?_d} be a sequence of nonnegative numbers and f(n) = Σ?_d, the sum being over divisors d of n. We say that f has the distribution function F if for all c ≥ 0, the number of integers n ≤ x for which f(n) > c is asymptotic to xF(c), and we investigate when F exists and when it is continuous.     

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.