Related papers: Partial Liouville sums over divisors

In this paper we give an additive representation of the factorial, which can be proven by a simple quick analytical argument. We also present some generalizations, which are linked, on the one hand to an arithmetical theorem proven by Euler…

Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…

We study large partial sums, localized with respect to the sums of variances, of a sequence of centered random variables. An application is given to the distribution of prime factors of typical integers.

We give exposition of a Liouville theorem established in \cite{Li3} which is a novel extension of the classical Liouville theorem for harmonic functions. To illustrate some ideas of the proof of the Liouville theorem, we present a new proof…

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann…

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.

We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight…

In the paper, we investigate the uniqueness problem of a power of an entire function that share one value partially with it's linear differential polynomial and obtain a result, which improves several previous results in a large scale. Also…

We introduce averaging operators on lattices $\mathbb{Z}^d$ and study the Liouville property for functions satisfying mean value properties associated to such operators. This framework encloses discrete harmonic, $p$-harmonic,…

We first prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial…

We prove an exponential integral estimate for the quadratic partial sums of multiple Fourier series on large sets that implies some new properties of Fourier series.

The problem of iterated partial summations is solved for some discrete distributions defined on discrete supports. The power method, usually used as a computational approach to finding matrix eigenvalues and eigenvectors, is in some cases…

In this paper we investigate some strong convergence theorems for partial sums with respect to Vilenkin system.

Euler discovered recurrence for divisor sum functions as a consequence of the pentagonal numbers theorem. With similar idea and also motivated by Ewell's work in 1977, we prove new recurrences for certain divisor sum functions and…

In this article, we consider an analogue of Arakelov theory of arithmetic surfaces over a trivially valued field. In particular, we establish an arithmetic Hilbert-Samuel theorem and studies the effectivity up to R-linear equivalence of…

In this paper, we apply the moving plane method to some degenerate elliptic equations to get a Liouville type theorem. As an application, we derive the a priori bounds for positive solutions of some semi-linear degenerate elliptic…

In this paper, we obtain bounds on the $L^1$ norm of the sum $\sum_{n\le x}\tau(n) e(\alpha n)$ where $\tau(n)$ is the divisor function.

We prove variational forms of the Barban-Davenport-Halberstam Theorem and the large sieve inequality. We apply our result to prove an estimate for the sum of the squares of prime differences, averaged over arithmetic progressions.

The number of tuples with positive integers pairwise relatively prime to each other with product at most $n$ is considered. A generalization of $\mu^{2}$ where $\mu$ is the M\"{o}bius function is used to formulate this divisor sum and…

Certain analytical expressions which "feel" the divisors of natural numbers are investigated. We show that these expressions encode to some extent the well-known algorithm of the sieve of Eratosthenes. Most part of the text is written in…