Related papers: On the mean values of some multiplicative function…
In this paper, we investigate the average behavior of ternary correlations for general $k$-divisor-bounded multiplicative functions, assuming certain second moment integral bounds for the associated $L$-functions. Our approach differs from…
In this article, we prove an asymptotic formula for mean values of long Dirichlet polynomials with higher order shifted divisor functions, assuming a smoothed additive divisor conjecture for higher order shifted divisor functions. As a…
For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.
The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and…
This short note provides a sharper upper bound of a well known inequality for the sum of divisors function. This is a problem in pure mathematics related to the distribution of prime numbers. Furthermore, the technique is completely…
For an odd integer $d > 1$ and a finite Galois extension $K/\mathbb{Q}$ of degree $d$, G. L\"{u} and Z. Yang \cite{lu3} obtained an asymptotic formula for the mean values of the divisor function for $K$ over square integers. In this…
Using estimates on Hooley's $\Delta$-function and a short interval version of the celebrated Dirichlet hyperbola principle, we derive an asymptotic formula for a class of arithmetic functions over short segments. Numerous examples are also…
Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and let $E(T)$ denote the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) := E(t) - 2\pi\Delta^*(t/(2\pi))$ with $\Delta^*(x)…
Given a multiplicative function f satisfying |f(n)| <= 1 for all n, the authors study the problem of obtaining explicit upper bounds on the mean-value 1/x |sum_{n <= x} f(n)|.
Some properties of integral averages of functions on intervals and their asymptotic behavior are investigated. The results are aimed at applications to entire and subharmonic functions.
We study sums with multiplicative functions that take values over a non-homogenous Beatty sequence. We then apply our result in a few special cases to obtain asymptotic formulas such as the number of integers in a Beatty sequence…
We provide uniform bounds on mean values of multiplicative functions under very general hypotheses, detecting certain power savings missed in known results in the literature. As an application, we study the distribution of the…
We discuss how one could study asymptotics of cyclotomic quantities via the mean values of certain multiplicative functions and their Dirichlet series using a theorem of Delange. We show how this could provide a new approach to Artin's…
The summation arithmetic functions with asymptotically independent summands are studied in the paper. We prove statements about the condition under which the summation arithmetic functions have asymptotically independent summands. It is…
We establish an asymptotic formula for the logarithmic mean value of a 1-bounded multiplicative function that is sharp in many cases of interest. We derive from it a variety of applications, making progress on several old problems. As a…
In this series, we investigate the calculation of mean values of derivatives of Dirichlet $L$-functions in function fields using the analogue of the approximate functional equation and the Riemann Hypothesis for curves over finite fields.…
Let $F({\bf x})={\bf x}^tQ_{\bf x}+\mathbf{b}^t{\bf x}+c\in\mathbb{Z}[{\bf x}]$ be a quadratic polynomial in $\ell (\ge 3 )$ variables ${\bf x} =(x_{1},...,x_{\ell})$, where $F({\bf x})$ is positive when ${\bf x}\in\mathbb{R}_{\ge…
We investigate the behavior of the divisor function in both short intervals and in arithmetic progressions. The latter problem was recently studied by \'E. Fouvry, S. Ganguly, E. Kowalski, and Ph. Michel. We prove a complementary result to…
The sum of proper divisors function $s(n)$ has been studied for more than 2000 years. In this paper we study statistical properties of the related function $S_s(n) := \sum_{d \mid n} s(d)$. This function arises from a generalization of the…
Asymptotic mean value properties, their converse and some other related results are considered for solutions to the $m$-dimensional Helmholtz equation (metaharmonic functions) and solutions to its modified counterpart (panharmonic…