Related papers: Averages of multiplicative functions along equidis…
In this paper we have proposed a median based estimator using known value of some population parameter(s) in simple random sampling. Various existing estimators are shown particular members of the proposed estimator. The bias and mean…
We continue investigations on the average number of representations of a large positive integer as a sum of given powers of prime numbers. The average is taken over a short interval, whose admissible length depends on whether or not we…
Let $r,\,f$ be multiplicative functions with $r\geqslant 0$, $f$ is complex valued, $|f|\leqslant r$, and $r$ satisfies some standard growth hypotheses. Let $x$ be large, and assume that, for some real number $\tau$, the quantities…
Roughly speaking, the spectrum of multiplicative functions is the set of all possible mean values. In this paper, we are interested in the spectra of multiplicative functions supported over powerful numbers. We prove that its real…
The problem of root mean square approximation of a square integrable function by finite linear combinations of exponential functions is considered. It is subdivided into linear and nonlinear parts. The linear approximation problem is…
Summation arithmetic functions with asymptotically independent terms are studied in the paper, the limit of which is the law of normal distribution. Assertions about the asymptotic behavior of the indicated functions are proved.
We evaluate friable averages of arithmetic functions whose Dirichlet series is analytically close to some complex power of the Riemann zeta function. We obtain asymptotic expansions resembling those provided by the Selberg-Delange method in…
We obtain a new upper bound for binary sums with multiplicative characters over variables belong to some sets, having small additive doubling.
We introduce methods to bound the mean of a discrete distribution (or finite population) based on sample data, for random variables with a known set of possible values. In particular, the methods can be applied to categorical data with…
Let $T$ be a general sampling statistic that can be written as a linear statistic plus an error term. Uniform and non-uniform Berry--Esseen type bounds for $T$ are obtained. The bounds are the best possible for many known statistics.…
We study generalized means whose domain may contain unbounded sets as well. We investigate usual properties of this type of means and also new attributes that regard for such means only. We examine how a mean defined on bounded sets can be…
We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.
We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…
By using the matrix formulation of the two-step approach to distributions of patterns in random sequences, recurrence and explicit formulas for the generating functions of successions in random permutations of arbitrary multisets are…
We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…
We give an asymptotic formula for correlations \[ \sum_{n\le x}f_1(P_1(n))f_2(P_2(n))\cdot \dots \cdot f_m(P_m(n))\] where $f\dots,f_m$ are bounded "pretentious" multiplicative functions, under certain natural hypotheses. We then deduce…
Set-valued quantiles for multivariate distributions with respect to a general convex cone are introduced which are based on a family of (univariate) distribution functions rather than on the joint distribution function. It is shown that…
A universally L^1 good sequence n_k is constructed with n_{k+1}-n_k tending to infinity. For ergodic transformations non-conventional ergodic averages of L^1 functions computed by using this sequence converge to the integral of the…
We give asymptotic estimates for the mean number of divisors of integers without small prime factors, integers with bounded ratios of consecutive divisors, and for practical numbers. In the last case, this confirms a conjecture of…
Some properties of the inverse of the Normal distribution are studied. Its derivatives, integrals and asymptotic behavior are presented.