Related papers: Fractional part correlations and the M\"obius func…

We investigate function field analogs of the distribution of primes, and prime $k$-tuples, in "very short intervals" of the form $I(f) := \{ f(x) + a : a \in \mathbb{F}_p \}$ for $f(x) \in \mathbb{F}_p[x]$ and $p$ prime, as well as…

We introduce and study the arithmetic function E_m(n), defined as the sum of the remainders of n when divided by the first m positive integers. Although the definition is elementary, the function encodes rich arithmetic structure. In this…

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which…

Apostol's Mobius functions of order k are generalized to depend on a second integer parameter m. Asymptotic formulas are obtained for the partial sums of these generalized functions.

Let $g:\mathbb{N}\to\{-1,1\}$ be a completely multiplicative function, $\mu$ be the M\"obius function and $\mu_2^2(n)$ be the indicator that $n$ is cubefree. We prove that $f=\mu^2g$ and $f=\mu_2^2g$ have unbounded partial sums. Our proofs…

The main objective of this article is to present $\nu$-fractional derivative $\mu$-differentiable functions by considering 4-parameters extended Mittag-Leffler function (MLF). We investigate that the new $\nu$-fractional derivative…

Motivated by recent results, we study sums of the form $S_f(x) = \sum_{n\leq x} f\left(\left\lfloor\frac{x}{n}\right\rfloor \right)$, where $f$ is an arithmetic function and $\left\lfloor\cdot\right\rfloor$ denotes the greatest integer…

The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as the summatory function of the classical M\"obius function. The Dirichlet inverse function $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shifted strongly…

In this paper we study correlation measures introduced in \cite{emme_asymptotic_2017}. Denote by $\mu_a(d)$ the asymptotic density of the set $\mathcal{E}_{a,d}=\{n \in \mathbb{N}, \ s_2(n+a)-s_2(n)=d\}$ (where $s_2$ is the sum-of-digits…

Sarnak's M\"{o}bius disjointness conjecture asserts that for any zero entropy dynamical system $(X,T)$, $\frac{1}{N} \sum_{n=1} ^N f(T^n x) \mu (n)= o(1)$ for every $f\in \mathcal{C}(X)$ and every $x\in X$. We construct examples showing…

Let X1, ..., Xn be arbitrary non-negative independent random variables with respective expected values $\mu_{i}$ at most one. We sketch but do not prove an equivalent conjecture to Feige's Conjecture $\mathbb{P} \left( \sum_{i=1}^{n} X_{i}…

Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $\mu$ and $\omega$ denote respectively the M\"obius function and the number of distinct prime factors function. We show that, given any set ${{\scr…

We study filters in the partition lattice formed by restricting to partitions by type. The M\"obius function is determined in terms of the easier-to-compute descent set statistics on permutations and the M\"obius function of filters in the…

This work, dealt with the classical mean value theorem and took advantage of it in the fractional calculus. The concept of a fractional critical point is introduced. Some sufficient conditions for the existence of a critical point is…

Let $\{b_H(t),t\in\mathbb{R}\}$ be the fractional Brownian motion with parameter $0<H<1$. When $1/2<H$, we consider diffusion equations of the type \[X(t)=c+\int_0^t\sigma\bigl(X(u)\bigr)\mathrm {d}b_H(u)+\int _0^t\mu\bigl(X(u)\bigr)\mathrm…

MacMahon showed that the generating function for partitions into at most $k$ parts can be decomposed into a partial fractions-type sum indexed by the partitions of $k$. In this present work, a generalization of MacMahon's result is given,…

We study the unweighted error-sum function $\mathcal{E}(x) \coloneqq \sum_{n \geq 0} ( x- p_n(x)/q_n(x) )$, where $p_n(x)/q_n(x)$ is the $n$th convergent of the continued fraction expansion of $x \in \mathbb{R}$. We prove that the Hausdorff…

Recently, Merca and Schmidt proved a number of identities relating partitions of an integer with two classic number-theoretic functions, namely the M\"obius function and Euler's totient function. Their demonstrations were mainly algebraic.…

Here we try and delienate the properties of the function that corresponds to fluctuations in the momentum distribution. The quantity denoted by $ N(k,k^{'}) $ is quite an interesting object. It satisfies various elegant sum rules and is…

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…