Related papers: A lower bound for the variance of generalized divi…

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…

Let $d_k(n)$ denote the $k$-fold divisor function. For a wide range of large $q$ the expected bound $$\sum_{n\leq x\atop {n\equiv a(q)}}d_k(n)-\text { main term }\approx \sqrt {x/q}$$ is shown to be true in an average sense -- for all $k$.…

In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar…

Let $\alpha$ be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions $f$ we show that the sum $\sum_{n \le x}\alpha(n) f(n)$, normalised to have mean square $1$, has a non-Gaussian…

For a wide class of sequences of integer domains $\mathcal{D}_n\subset\mathbb{N}^d$, $n\in\mathbb{N}$, we prove distributional limit theorems for $F(X_1^{(n)},\ldots,X_d^{(n)})$, where $F$ is a multivariate multiplicative function and…

Iannucci considered the positive divisors of a natural number $n$ that do not exceed the square root of $n$ and found all numbers whose such divisors are in arithmetic progression. Continuing the work, we define large divisors to be…

The purpose of this paper is to introduce the notion of a generalized derivation which derivates a prescribed family of smooth vector-valued functions of several variables. The basic calculus rules are established and then a result derived…

In this paper, we consider the general divisor functions over Piatetski-Shapiro sequences. We can give some general results which contain some special divisor functions. Precisely, we extend the divisor problem over Piatetski-Shapiro…

In [arXiv:2212.04969], the authors stated some conjectures on the variance of certain sums of the divisor function $d_k(n)$ over number fields, which were inspired by analogous results over function fields proven in [arXiv:2107.01437].…

We prove new exact formulas for the generalized sum-of-divisors functions, $\sigma_{\alpha}(x) := \sum_{d|x} d^{\alpha}$. The formulas for $\sigma_{\alpha}(x)$ when $\alpha \in \mathbb{C}$ is fixed and $x \geq 1$ involves a finite sum over…

For a general free L\'evy process, we prove the existence of its higher variation processes as limits in distribution, and identify the limits in terms of the L\'evy-It\^o representation of the original process. For a general free compound…

Let $X$ be a random variable with distribution function $F,$ and $X_{1},X_{2},...,X_{n}$ are independent copies of $X.$ Consider the order statistics $X_{i:n},$ $i=1,2,...,n$ and denote $F_{i:n}(x)=P\{X_{i:n}\leq x\}.$ Using majorization…

Although there is an extensive literature on the upper bound for cumulative standard normal distribution, there are relatively not sharp for all values of the interested argument x. The aim of this paper is to establish a sharp upper bound…

We provide new upper bounds for sums of certain arithmetic functions in many variables at polynomial arguments and, exploiting recent progress on the mean-value of the Erd\H os-Hooley $\Delta$-function, we derive lower bounds for the…

We study averages along the integers using the divisor function $d(n)$, and defined as $$K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) \,f(x+n) , $$ where $D(N) = \sum _{n=1} ^N d(n) $. We shall show that these averages satisfy a…

We prove an averaged version of a claim suspected to be true by Alladi, Erd\"os, and Vaaler. Qualitatively, the result states that a divisor sum of a multiplicative function, which obeys certain size constraints, derives most of its value…

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…

Hal\'asz's Theorem gives an upper bound for the mean value of a multiplicative function $f$. The bound is sharp for general such $f$, and, in particular, it implies that a multiplicative function with $|f(n)|\le 1$ has either mean value…

We prove an asymptotic formula for the shifted convolution of the divisor functions $d_k(n)$ and $d(n)$ with $k \geq 4$, which is uniform in the shift parameter and which has a power-saving error term, improving results obtained previously…

We study the triple convolution sum of the generalised divisor functions $$\sum_{n\leq x} d_k(n+h)d_l(n)d_m(n-h),$$ where $h \le x^{1-\epsilon}$ for any $\epsilon>0$ and $d_k(n)$ denotes the generalised divisor function which counts the…