Related papers: On the mean values of some multiplicative function…

In this paper we prove the mean values of some multiplicative functions connected with the divisor function on the short interval of summation.

In this paper, we shall establish a rather general asymptotic formula in short intervals for a classe of arithmetic functions and announce two applications about the distribution of divisors of square-full numbers and integers representable…

A classical problem in number theory is showing that the mean value of an arithmetic function is asymptotic to its mean value over a short interval or over an arithmetic progression, with the interval as short as possible or the modulus as…

We consider random multiplicative functions taking the values $\pm 1$. Using Stein's method for normal approximation, we prove a central limit theorem for the sum of such multiplicative functions in appropriate short intervals.

The paper considers estimates for the asymptotics of summation functions of bounded multiplicative arithmetic functions. Several assertions on this subject are proved and examples are considered.

Using recent results from the theory of integer points close to smooth curves, we give an asymptotic formula for the distribution of values of a class of integer-valued prime-independent multiplicative functions.

We introduce a general result relating "short averages" of a multiplicative function to "long averages" which are well understood. This result has several consequences. First, for the M\"obius function we show that there are cancellations…

We study a general class of multiplicative functions by relating "short averages" to its "long average". More precisely, we estimate asymptotically the variance of such a class of functions in short intervals using Fourier analysis and…

Asymptotic properties of certain arithmetic functions involving exponential divisors are investigated.

We extend the Matom\"{a}ki-Radziwi\l\l{} theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a…

In this article, we prove an asymptotic formula for the mean value of long smoothed Dirichlet polynomials with divisor coefficients. Our result has a main term that includes all lower order terms and a power saving error term. This is…

The primary objective of this paper is to employ methods from analytic number theory to investigate the mean value properties of a composite function involving the Dirichlet divisor function and a generalized minimal power function.…

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…

We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors' new proof of Halasz's theorem on mean values to this simpler setting. Several of the technical difficulties that arise over…

We study the symmetry in short intervals of arithmetic functions with non-negative exponential sums.

A celebrated result of Hal\'asz describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions…

We exploit the properties of a sequence of functions that approximate the divisor functions and combine them with an analytical formula of a delta-like sequence to give a new proof of a theorem of Gronwall on the asymptotic of the divisor…

We use multiple zeta functions to prove, under suitable assumptions, precise asymptotic formulas for the averages of multivariable multiplicative functions. As applications, we prove some conjectures on the average number of cyclic…

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 establish asymptotic formulae for various correlations involving general divisor functions $d_k(n)$ and partial divisor functions $d_l(n,A)=\sum_{q|n:q\leq n^A}d_{l-1}(q)$, where $A\in[0,1]$ is a parameter and $k,l\in\mathbb{N}$ are…